We provide sequences and series practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on sequences and series skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.

#### List of sequences and series Questions

Question No | Questions | Class |
---|---|---|

1 | If ( m ) th term of an A.P. is ( n ) and ( n ) th term is ( m, ) then the ( (m+n) ) th term is ( mathbf{A} cdot mathbf{0} ) B. ( m+n-1 ) ( mathrm{c} cdot m+n ) D. ( frac{m n}{m+n} ) | 11 |

2 | ( ln ) an A.P., if ( m^{t h} ) term is ( n ) and the ( n^{t h} ) term is ( mathrm{m}, ) where ( boldsymbol{m} neq boldsymbol{n} ) find the ( boldsymbol{p}^{t h} ) term. A ( . m+n+p ) в. ( m-n+p ) c. ( m+n-p ) ( mathbf{D} cdot m-n-p ) | 11 |

3 | Find the arithmetic mean of first 10 natural numbers. A . 55 в. 550 ( c .5 .5 ) D. None of the above | 11 |

4 | Prove that ( frac{C_{1}}{C_{0}}+frac{2 C_{2}}{C_{1}}+frac{3 C_{3}}{C_{2}}+ldots+frac{n cdot C_{n}}{C_{n-1}}= ) ( frac{n(n+1)}{2} ) | 11 |

5 | If ( n ) arithmetic means are inserted between 1 and 31 such that ratio of first and ( n^{t h} ) mean is ( 3: 29, ) then what is the value of ( n ? ) A . 10 B. 14 c. 18 D. 23 | 11 |

6 | In a geometric progression with common ratio ‘q’, the sum of the first 109 terms exceeds the sum of the first 100 terms by ( 12 . ) If the sum of the first nine terms of the progression is ( frac{boldsymbol{lambda}}{boldsymbol{q}^{100}} ) then the value of ( lambda ) equals to A . 10 B. 14 c. 12 D. 22 | 11 |

7 | if ( 1+x^{2}=sqrt{3} x, ) then ( , prod_{n=1}^{24}left(x^{n}+frac{1}{x^{n}}right) ) is equal to: | 11 |

8 | In a sequence, ( a_{n}=n^{2}-1 ) then ( a_{n+1} ) is equal to A ( cdot a^{2}-5 n ) B . ( n^{2}-2 n ) c. ( a^{2}+10 n ) D. ( n^{2}+2 n ) | 11 |

9 | Let f(x) be a polynomial function of second degree. f(1)=f(-1) and a, b,c are in A. P, then f'(a), S'(b), f ‘c) are in [2003] (a) Arithmetic -Geometric Progression (b) A.P (c) GP (d) H.P. | 11 |

10 | The first and the last term of an AP are 17 and 350 respectively. If the common difference is ( 9, ) how many terms are there and what is the sum? | 11 |

11 | The first three of four given numbers are in G.P. and last three are in A.P. whose common difference is ( 6 . ) If the first and last numbers are same, then first will be? A . 2 B. 4 ( c cdot 6 ) D. 8 | 11 |

12 | Fifth term of a GP is 2, then the product of its 9 terms is [2002] (a) 256 (b) 512 (c) 1024 (d) none of these | 11 |

13 | Find the sum of the first 25 terms of an A.P whose ( n ) th term is given by ( a_{n}= ) ( 2-3 n ) ( mathbf{A} cdot-925 ) B . -928 ( mathbf{c} .-923 ) D ( .-929 ) | 11 |

14 | Find out which of the following sequences are arithmetic progressions. For those which are arithmetic progressions, find out the common difference. ( 3,3,3,3, dots . ) | 11 |

15 | Find the total area of 14 squares whose sides are ( 11 mathrm{cm}, 12 mathrm{cm}, ldots, 24 mathrm{cm} ) respectively | 11 |

16 | Find the sum of the series: ( mathbf{5}+mathbf{1 3}+mathbf{2 1}+dots+mathbf{1 8 1} ) | 11 |

17 | A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows ( R s 200 ) for the first day, ( boldsymbol{R} boldsymbol{s} ) 250 for the second day Rs 300 for the third day, etc. the penalty for each succeeding day being ( R s 50 ) more than for the preceding day. How much money the contractor has to pay as penalty, if he has delayed the work by 30 days? | 11 |

18 | find the sum : tan ( x ) tan ( 2 x+ ) ( tan 2 x tan 3 x+dots dots dots+dots ) ( tan n x tan (n+1) x ) | 11 |

19 | 16.2,65,7,59,12,53,。 (1) 15, 42 2 17, 45 (3) 17,47 (4) 18.48 | 11 |

20 | Sum of nine terms of the series ( log a+ ) ( log frac{a^{2}}{b}+log frac{a^{3}}{b^{2}}+log frac{a^{4}}{b^{3}}+ldots . . i s- ) A . ( 36 log a-28 log b ) B. ( 45 log a-36 log b ) c. ( 55 log a-45 log b ) D. ( 66 log a-55 log b ) | 11 |

21 | The middle terms, if four different numbers are in proportion are called A. Antecedents B. Means C. Extremes D. consequents | 11 |

22 | The fourth term of an A.P. is 11 and the eighth term exceeds twice the fourth term by 5. Find the A.P. and the sum of first ( mathbf{5 0} ) terms. A . 3540 в. 6754 c. 4850 D. 7819 | 11 |

23 | Which term of the sequence ( mathbf{3}, mathbf{8}, mathbf{1 3}, mathbf{1 8}, dots dots . . ) is ( mathbf{4 9 8} ) A. 95 th B. 100 th c. ( 102 t h ) D. ( 101 t h ) | 11 |

24 | Find the 12 th term of a G.P whose 8 th term is 192 and the common ratio is 2 | 11 |

25 | ( mathbf{2} cdot mathbf{1}^{2}+mathbf{3} cdot mathbf{2}^{2}+mathbf{4} cdot mathbf{3}^{2}+ldots ) up to ( boldsymbol{n} ) terms ( = ) A ( cdot frac{n(n+1)(n+2)(3 n+1)}{12} ) B ( cdot frac{n(n+2)(n+3)(n+11)}{12} ) c. ( frac{n(n+1)(n+2)(3 n-2)}{6} ) D. ( frac{n(n+2)(n+5)(n+8)}{6} ) | 11 |

26 | If the term ( 10, k,-2 ) are in A.P then the value of ( k ) is ( mathbf{A} cdot mathbf{1} ) B. 3 ( c cdot 2 ) D. | 11 |

27 | If the set natural numbers, is partitioned into subsets ( boldsymbol{S}_{1}= ) ( {1}, S_{2}={4,5,6}, S_{4}={7,8,9,10} ) The last terms of these groups is ( 1,1+ ) ( mathbf{2}, mathbf{1}+mathbf{2}+mathbf{3}, mathbf{1}+mathbf{2}+mathbf{3}+mathbf{4} ldots ldots . ) Find the sum of the elements in the subset ( S_{50} ) | 11 |

28 | The sum of 1 st ( n ) terms of the series ( frac{1^{2}}{1}+frac{1^{2}+2^{2}}{1+2}+frac{1^{2}+2^{2}+3^{2}}{1+2+3}+dots dots ) ( ^{text {A } cdot frac{n+2}{3}} ) в. ( frac{n(n+2)}{3} ) c. ( frac{n(n-2)}{3} ) D. ( frac{n(n-2)}{6} ) | 11 |

29 | Find the middle term of the sequence formed by all three-digit numbers which leave a remainder 5 when divided by 7 Also find the sum of all numbers on both sides of middle term. | 11 |

30 | Show that the ratio of the sum of first ( n ) terms of a G.P. to the sum of terms form ( (n+1)^{t h} ) to ( (2 n)^{t h} ) term is ( frac{1}{r^{n}} ) | 11 |

31 | ( ln a ) G. P. of even number of terms the sum of all terms is 5 times the sum of the odd term. Find the common ratio of the G.P. A .2 B. 3 ( c cdot 4 ) D. 5 | 11 |

32 | Let ( boldsymbol{A}=mathbf{1 6}-mathbf{4}+mathbf{2}^{-mathbf{4}}+mathbf{3}^{-mathbf{4}}+mathbf{4}^{-mathbf{4}}+dots ) and ( boldsymbol{B}=mathbf{1}^{-mathbf{4}}+mathbf{3}^{-mathbf{4}}+mathbf{5}^{-mathbf{4}}+mathbf{7}^{-mathbf{4}}+ldots ) The ratio ( frac{A}{B} ) in the lowest form is A ( cdot frac{16}{15} ) в. ( frac{15}{14} ) c. ( frac{15}{16} ) D. ( frac{13}{12} ) | 11 |

33 | Determine the sum of the arithmetic series ( 10+20+30+ldots 600 ) A . 13500 B. 15500 c. 18300 D. 19100 | 11 |

34 | Find ( k ) if ( k, 12,24, dots ) are in GP ( A cdot 6 ) B. 3 ( c cdot 9 ) D. none of these | 11 |

35 | Fill up the gaps (shown by-) in the following A.P. ( 34,-,-,-,-, 48 ) | 11 |

36 | If the ratio of the ( 11^{t h} ) term of an A.P. to its ( 18^{t h} ) term is ( 2: 3, ) find the ratio of the sum of first five terms to the sum of its first 10 terms | 11 |

37 | Find the common ratio in the following G.P. ( -5,1, frac{-1}{5}, dots ) | 11 |

38 | Assertion STATEMENT -1: If ( x, y, z ) are the sides of a triangle such that ( x+y+z=1 ), then ( left[frac{2 x-1+2 y-1+2 z-1}{3}right] geq((2 x- ) 1) ( (2 y-1)(2 z-1))^{1 / 3} ) Reason STATEMENT-2: For positive numbers their A.M., G.M. and H.M. satisfy the relation ( boldsymbol{A} . boldsymbol{M} .>boldsymbol{G} . boldsymbol{M} .>boldsymbol{H} . boldsymbol{M} ) A. Statement-1 is True, Statement-2 is True; Statement -2 is a correct explanation for Statement- – B. Statement-1 is True, Statement-2 is True ; Statement-2 is NOT a correct explanation for Statement- c. Statement- – is True, Statement- -2 is False D. Statement-1 is False, Statement-2 is True | 11 |

39 | Find the average of first 20 multiples of 7 | 11 |

40 | Find the ( n^{t h} ) term of ( 2,4,6,8,10, dots dots dots ) | 11 |

41 | Find the 12 th term from the end of the following arithmetic progressions. ( mathbf{1}, mathbf{4}, mathbf{7}, mathbf{1 0}, dots dots dots, mathbf{8 8} ) | 11 |

42 | The sum of first 9 terms of an AP is 81 and the sum of its first 20 terms is 400 . Find the first term and the common difference of the AP. | 11 |

43 | The sum of the series ( frac{5}{13}+frac{55}{13^{2}}+frac{555}{13^{3}}+ ) ( dots dots infty ) is ( mathbf{A} cdot frac{65}{36} ) B. ( frac{65}{32} ) ( mathbf{C} cdot frac{25}{36} ) D. none of these | 11 |

44 | Sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio ( (3+2 sqrt{2}):(3-2 sqrt{2}) ) | 11 |

45 | f ( a, b, c ) be positive and ( a b(a+b)+ ) ( boldsymbol{b} boldsymbol{c}(boldsymbol{b}+boldsymbol{c})+boldsymbol{c} boldsymbol{a}(boldsymbol{c}+boldsymbol{a}) geq boldsymbol{lambda} boldsymbol{a} boldsymbol{b} boldsymbol{c}, ) then value of ( lambda ) is | 11 |

46 | ( ln operatorname{an} A cdot P . S_{3}=6, S_{6}=3, ) then it’s common difference is equal to? ( A cdot 3 ) B. – 1 ( c . ) D. None of these | 11 |

47 | The ( r^{t h}, s^{t h} ) and ( t^{t h} ) terms of a certain ( G . P ) are ( R, S ) and ( T ) respectively. Prove that ( boldsymbol{R}^{s-t} boldsymbol{S}^{t-r} boldsymbol{T}^{r-s}=mathbf{1} ) | 11 |

48 | The sum of terms upto 10 terms of series ( 12,15,18,21,24,27, dots dots ) | 11 |

49 | If ( 11 A M . s ) are inserated between 28 and ( 10, ) then the middle term in the series is A . 15 B. 19 ( c cdot 21 ) D. None of these | 11 |

50 | If sum of ( n ) terms of a sequence is given by ( S_{n}=2 n^{2}+3 n, ) find its ( 50^{t h} ) term. | 11 |

51 | Form an arithmetic progression with its first term as 14 and common difference as -3 | 11 |

52 | Sn denotes the sum of first ( n ) terms of the A.P.1 ( , 2,3,4, dots . ., ) then ( S_{2 n}=3 S n ) ( frac{boldsymbol{S}_{boldsymbol{3 n}}}{boldsymbol{S} boldsymbol{n}}=? ) | 11 |

53 | The ratio of sum of ( m ) and ( n ) terms of an A.P is ( m^{2}: n^{2}, ) then the ratio of ( m^{t h} ) and ( n^{t h} ) term. | 11 |

54 | If three positive real numbers ( x, y, z ) satisfy ( boldsymbol{y}-boldsymbol{x}=boldsymbol{z}-boldsymbol{y} ) and ( boldsymbol{x} boldsymbol{y} boldsymbol{z}=boldsymbol{4} ) then what is the minimum possible value of ( y ? ) ( A cdot 2^{frac{1}{5}} ) B. ( 2^{frac{2}{5}} ) ( c cdot 2^{frac{1}{4}} ) ( D cdot 2^{frac{3}{7}} ) | 11 |

55 | Consider the arithmetic sequence 9,15 21 a) Write the algebraic form of this sequence. b) Find the twenty fifth term of this sequence. c) Find the sum of terms from twenty fifth to fiftieth of this sequence. d) Can the sum of some terms of this sequence be ( 2015 ? ) Why? | 11 |

56 | Calculate geometric mean for the following values 1,3,5,7,10,12 A. 1.82386 в. 2.82386 ( c .3 .82386 ) D. 4.82386 | 11 |

57 | The H.M. of two number is 4 and their ( A ) ( m . ) and ( G . M . ) satisfy the relation ( 2 A+ ) ( G^{2}=27, ) then the numbers are : | 11 |

58 | Which term of the sequence ( 72,70,68,66, dots ) is ( 40 ? ) | 11 |

59 | If the ( 4^{t h} ) and ( 7^{t h} ) term of a G.P. are 54 and 1458 respectively, then find its ¡) Common ratio ii) ( 6^{t h} ) term | 11 |

60 | The terms of an infinitely decreasing G.P. in which all the terms are positive, the first term is 4 and the difference between the third and fifth term is ( frac{32}{81} ) then find the common ratio. | 11 |

61 | The common difference of ( -2,-4,-6,-8, dots dots dots . . ) is : A . -2 B. – ( c cdot 2 ) ( D ) | 11 |

62 | The digits of a positive integer, having three digits are in A.P. and their sum is 15. The number obtained by reversing the digits is 594 less than the original number. Find the number. | 11 |

63 | If ( I, m, n ) are the direction cosines of ( a ) line OP then the maximum value of I.m.n is A ( cdot frac{1}{sqrt{3}} ) в. ( frac{1}{3 sqrt{3}} ) ( c cdot frac{1}{3} ) D. | 11 |

64 | If ( a, b, c ) are in A.P., then the following are also in A.P. ( frac{1}{b c}, frac{1}{c a}, frac{1}{a b} ) A . True B. False | 11 |

65 | Check if 0 is a term of the ( A P: 31,28,25 ) | 11 |

66 | The value of the sum ( 1.2 .3+2.3 .4+ ) ( mathbf{3} . mathbf{4 . 5}+ldots ldots . ) upto ( n ) terms ( = ) A ( cdot frac{1}{6} n^{2}left(2 n^{2}+1right) ) B – ( frac{1}{6} n^{2}left(n^{2}-1right)(2 n-1)(2 n+3) ) C ( cdot frac{1}{8}left(n^{2}+1right)left(n^{2}+5right) ) D・去 ( (n)(n+1)(n+2)(n+3) ) | 11 |

67 | If ( a, b, c ) and ( d ) are in ( G P, ) then ( (a+b+ ) ( boldsymbol{c}+boldsymbol{d})^{2} ) is equal to: ( mathbf{A} cdot(a+b)^{2}+(c+d)^{2}+2(b+c)^{2} ) B. ( (a+b)^{2}+(c+d)^{2}+2(a+c)^{2} ) ( mathbf{C} cdot(a+b)^{2}+(c+d)^{2}+2(b+d)^{2} ) D. ( (a+b)^{2}+(c+d)^{2}+(b+c)^{2} ) E ( cdot(a+b)^{2}+(c+d)^{2}+(b-c)^{2} ) | 11 |

68 | If ( a, b, c ) are in ( A . P ., alpha, beta, gamma ) in H.P., ( boldsymbol{a} boldsymbol{alpha}, boldsymbol{b} boldsymbol{beta}, boldsymbol{c} gamma ) in G.P. (with common ratio not equal to ( 1 . ) ), then prov that ( a: b: c= ) ( frac{1}{gamma}: frac{1}{beta}: frac{1}{alpha} ) | 11 |

69 | If the ( p^{t h}, q^{t h} ) and ( r^{t h} ) term of an arithmetic sequence are ( a, b ) and ( c ) respectively, then the value of ( [boldsymbol{a}(boldsymbol{q} boldsymbol{r})+boldsymbol{b}(boldsymbol{r} boldsymbol{p})+boldsymbol{c}(boldsymbol{p} boldsymbol{q})]= ) ( A ) B. – ( c cdot c ) D. ( 1 / 2 ) | 11 |

70 | The minimum value of ( 4 e^{x}+9 e^{-x} ) is A. 5 B. 25 ( c cdot 12 ) D. 13 | 11 |

71 | Find the common difference of ( 4, frac{15}{2}, 11 ) ( A cdot frac{7}{2} ) B. c. ( frac{11}{41} ) D. ( frac{41}{11} ) | 11 |

72 | Add the following: ( (-100)+(-92)+(-84)+ldots . .+92 ) | 11 |

73 | Find the sum to the series ( 1 . n+2(n- ) 1) ( +3(n-2)+ldots+n .1 ) | 11 |

74 | f ( x, y, z ) are positive then minimum value of ( boldsymbol{x}^{log y-log z}+boldsymbol{y}^{log z-log x}+boldsymbol{z}^{log boldsymbol{x}-log boldsymbol{y}} ) is ( A cdot 3 ) B. c. 9 D. 16 | 11 |

75 | In following symbol series, some of the symbols are missing which are given in that order as one of the alternatives below it. Choose the correct alternative. ( $ $ operatorname{ltex} tleft{x+_{-} x x_{-}+++x x_{-}+++xright} $ $ ) ( A cdot x++x ) B. ( +x x+ ) ( c cdot x x x+ ) D. ( x+x+ ) ( E cdot x+x x ) | 11 |

76 | If ( n^{t h} ) term of an A.P. is ( 2 n+1 ), then find its common difference. | 11 |

77 | For the given A.P. 5,10,15,20 Find the common difference (d). | 11 |

78 | The sum to infinite of the series ( mathbf{1}+frac{2}{3}+frac{6}{3^{2}}+frac{10}{3^{3}}+frac{14}{3^{4}}+dots dots ) ( A cdot 2 ) B. 3 ( c cdot 4 ) D. 6 | 11 |

79 | The quantities ( frac{1}{log _{4} 3}, log _{3} 8, frac{1}{log _{16} 3} ) are in A. A.P. в. G.P. c. н.P D. None of these | 11 |

80 | Which term of the ( boldsymbol{A} . boldsymbol{P} mathbf{2 5}, mathbf{2 0}, mathbf{1 5} dots dots ) is the first ( – ) ve term. | 11 |

81 | Prove that no matter what the real number ( a ) and ( b ) are, the sequence with the ( n t h ) term ( (a+n b) ) is always an ( A P ) Also find the sum of first 20 terms. | 11 |

82 | Divide 20 into four parts which are in arithmetic progression such that the product of the first and fourth is to the product of the second and third is in the ratio 2: 3 then least value of them is A .2 B. 4 ( c cdot 6 ) D. 8 | 11 |

83 | Sum to ( n ) terms the series: ( mathbf{1} times mathbf{3}+mathbf{3} times mathbf{5}+mathbf{5} times mathbf{7}+mathbf{7} times mathbf{9}+dots ) | 11 |

84 | How many arithmetic progressions with 10 terms are there, whose first term is in the set {1,2,3,4} and whose common difference is in the set {3,4,5,6,7}( ? ) | 11 |

85 | Find the sum given. ( 34+32+30+ldots+10 ) | 11 |

86 | Find ( x, ) if the given numbers are in A.P. ( (a+b)^{2}, x,(a-b)^{2} ) | 11 |

87 | Find the sum of the number of terms in the geometric progressions in ( 0.15,0.015,0.0015, dots 20 ) | 11 |

88 | 15. Let a, a, …, an be positive real numbers in geometric progression. For each n, let An, G, H, be respectively, the arithmetic mean, geometric mean, and harmonic mean of a,,a,, …, ap. Find an expression for the geometric mean of G,G2, …, G, in terms of A1, A2, …, An, H1,H2, …,Hn. (2001 – 5 Marks) | 11 |

89 | Find the common difference and write the next three terms of the A.P. ( mathbf{3},-mathbf{2},-mathbf{7},-mathbf{1 2}, dots ) | 11 |

90 | ( mathbf{3 5}, mathbf{4 1}, mathbf{4 7}, mathbf{5 3}, mathbf{5 9}, dots dots ) For this sequence, write down (a) the next term (b) the ( n ) th term | 11 |

91 | Find the ( 7^{t h} ) term from the last in the G.P ( 2,4,8,16, dots 60 ) terms | 11 |

92 | The sum of the ( 4^{t h} ) and ( 8^{t h} ) terms of an AP is 24 and the sum of the ( 6^{t h} ) and ( 10^{t h} ) terms is ( 44 . ) Find the first three terms of the AP. | 11 |

93 | Find the common ratio and the general term of the following geometric sequences. ( mathbf{0 . 0 2}, mathbf{0 . 0 0 6 , 0 . 0 0 1 8}, ldots . ) | 11 |

94 | 21. A person is to count 4500 currency notes. Let a, denote the number of notes he counts in the nth minute. Ifa, = a, = … = 010 = 150 and a 10, 0, 1, … are in an AP with common difference -2, then the time taken by him to count all notes is [2010 (a) 34 minutes (b) 125 minutes (c) 135 minutes (d) 24 minutes | 11 |

95 | 11. The sum of series + – + 2! 4! + ….. is 6! (e² – 2) (e2 – 1) (d) (e²-1) 2e. 2 – . | 11 |

96 | Find the sum of the series ( 1+(1+ ) ( boldsymbol{x})+left(mathbf{1}+boldsymbol{x}+boldsymbol{x}^{2}right)+ldots ) to ( n ) terms, ( boldsymbol{x} neq mathbf{1} ) | 11 |

97 | If the angles A, B and C of a triangle are in an arithmeti progression and if a, b and c denote the lengths of the side opposite to A, B and C respectively, then the value of the (2010) expressionsin 2C+sin 2A is (a) () © 1 (a) və | 11 |

98 | If the sum of three consecutive terms of an increasing A.P. is 51 and the product of the first and third of these terms is ( 273, ) then the third term is A . 13 B. 9 ( c cdot 21 ) D. 17 | 11 |

99 | 19. The first two terms of a geometric progression add up to 12. the sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term is [2008] (a) 4 (6) – 12 (c) 12 (d) 4 | 11 |

100 | Solve for a ( frac{3 a-2}{7}-frac{a-2}{4}=2 ) | 11 |

101 | A progression of the form ( a, a r, a r^{2}, dots ) is a A . geometric series B. harmonic series c. arithmetic progression D. geometric progression | 11 |

102 | The sum of ( n ) terms of the ( G cdot P .3,6,12, ldots ) is ( 381 . ) Find the value of ( boldsymbol{n} ) | 11 |

103 | Prove that the product of ( n ) geometric mean between any two numbers is ( n ) th power of their ( G . M ) | 11 |

104 | The sum of the first ( n ) terms of an A.P. is half of the sum of the next ( n ) terms. In the usual notation, the value ( frac{boldsymbol{S}_{3 n}}{boldsymbol{S}_{n}} ) is A . 10 B. 8 ( c cdot 6 ) D. | 11 |

105 | Find the sum of the following arithmetic series: ( 2+9+16+23+30+ ) ( ldots . . ) to 20 terms. A. 1310 в. 1340 c. 1370 D. 1350 | 11 |

106 | If ( 4 A M^{prime} s ) are inserted between ( frac{1}{2} ) and 3 then third ( A M ) is A . -2 B. 2 ( c cdot-1 ) D. | 11 |

107 | If ( sin x+cos x=sqrt{y+frac{1}{y}}, x epsilon[0, pi] ) and ( boldsymbol{y}>0, ) then A . ( x=pi / 4 ) B. ( x=frac{pi}{2} ) c. ( x=frac{pi}{6} ) D. ( x=3 pi / 4 ) | 11 |

108 | 5. Ifx>1, y>1, z>1 are in GP., then 1 + In x In r’ 1+ In y 1- 1+ In y’1+ In z (1998 – 2 Marks) are in (a) A.P. (b) H.P. (c) GP Foro nositive intacar n lat (d) None of these | 11 |

109 | ( ln Delta A B C ) ( a^{2}, b^{2}, c^{2} ) are in A.P. Show cot ( A, cot B, cot C ) are in A.P | 11 |

110 | ( boldsymbol{x}=mathbf{1}+frac{mathbf{1}}{mathbf{2} times underline{mathbf{1}}}+frac{mathbf{1}}{mathbf{4} times underline{mathbf{2}}}+frac{mathbf{1}}{mathbf{8} times underline{mathbf{3}}} ) ( mathbf{A} cdot e^{1 / 2} ) B ( cdot e^{2} ) ( c ) ( D cdot frac{1}{1} ) | 11 |

111 | Minimum value of ( lambda ) for which the equation ( 9 a^{2 x}-(lambda+2) a^{x}+4=0 ) ( a>1 ) has atleast one real solution is: A .4 B. 6 c. 8 D. 10 | 11 |

112 | ( mathbf{1}-mathbf{1}+mathbf{1}-mathbf{1}+mathbf{1}-mathbf{1}+ ) ( dots dots dots(mathbf{1} mathbf{0} mathbf{1} t i m e s)= ) | 11 |

113 | If the ( A . M ) is twice the ( G . M . ) of the numbers ( a ) and ( b ), then ( a: b ) will be This question has multiple correct options A ( cdot frac{2-sqrt{3}}{2+sqrt{3}} ) B. ( frac{2+sqrt{3}}{2-sqrt{3}} ) c. ( frac{sqrt{3}-2}{sqrt{3}+2} ) D. ( frac{sqrt{3}+2}{sqrt{3}-2} ) | 11 |

114 | Chose the correct alternative from the given below questions: The sequency -25,-23,-21,-19 A. is an ( A ).P. Reason ( d=3 ) B. is an A.P. Reason ( d=2 ) c. is an A.P. Reason ( d=4 ) D. is not an ( A . P ) | 11 |

115 | If ( n^{t h} ) term of ( A P ) is ( 4 n+1 ), then ( A M ) of ( 11^{t h} ) to ( 20^{t h} ) terms is A . 61.5 B. 63 ( c .63 .5 ) D. 62 | 11 |

116 | The sum of first ( ^{prime} n^{prime} ) terms of an Arithmetic Progression is ” ( 5 n^{2}-2 n ” ) Find the ( 20^{t h} ) term? A . 1960 в. 183 c. 203 D. 193 | 11 |

117 | The sum of first ( p ) terms of an A.P. is equal to the sum of the first ( q ) terms, then find the sum of the first ( (p+q) ) terms. | 11 |

118 | The value of ( x ) if ( 4,6, x ) are in GP A. 9 B. 5 ( c . pi ) D. ( frac{22}{7} ) | 11 |

119 | The value of ( frac{1}{97}+frac{2}{97}+ldots . .+frac{96}{97} ) is A . 48 в. -48 ( c cdot 1 ) D. None of the above | 11 |

120 | f ( a, b, c ) are positive real numbers, prove that ( frac{b^{2}+c^{2}}{b+c}+frac{c^{2}+a^{2}}{c+a}+frac{a^{2}+b^{2}}{a+b} geq a+ ) ( boldsymbol{b}+boldsymbol{c} ) | 11 |

121 | ( f t_{n}=3 n+5, ) then find A.P. | 11 |

122 | ( 1-6+36-216+ldots . . ) is a geometric sequence, find ( r ) ( mathbf{A} cdot mathbf{1} ) B. – 6 ( c .36 ) D . -216 | 11 |

123 | If ( frac{b+c-a}{a}, frac{c+a-b}{b}, frac{a+b-c}{c} ) are in ( A . P ., ) then which of the following is in A.P.? This question has multiple correct options A. ( a, b, c ) B ( cdot a^{2}, b^{2}, c^{2} ) c. ( frac{1}{a}, frac{1}{b}, frac{1}{c} ) D. ( b c, a c, a b ) | 11 |

124 | (1991- 4 Marks) The real numbers X1, X2, X3 satisfying the equation 372 + Bx+y=0 are in AP. Find the intervals in which B and y lie. (1996 – 3 Marks) | 11 |

125 | What is the sum of all prime numbers between 100 and ( 120 ? ) ( mathbf{A} cdot 652 ) в. 650 ( c cdot 644 ) D. 533 | 11 |

126 | Prove: ( 1+2+3+4+5 cdot . .+n= ) ( frac{n(n+1)}{2} ) | 11 |

127 | Identify the correct sequence represents a infinite geometric sequence. A .3,6,12,24,48 B. ( 1+2+4+8+ldots ) c. 1,-1,1,-1,1 D. ( 1,3,4,5,6 dots ) | 11 |

128 | Find the ( 10^{t h} ) term of A.P whose sum of ( n ) terms is given by ( 2 n^{2}+3 n . ) Also find the ( n^{t h} ) term. | 11 |

129 | ( 4,9,25, ?, 121,169 ) A . 36 B . 49 ( c .64 ) D. 81 | 11 |

130 | The geometric mean of the first n terms of the series ( a, a r, a r^{2}, dots, ) is A ( cdot a r^{n / 2} ) 2 B ( . a r^{n} ) c. ( a r^{(n-1) / 2} ) D. ( a r^{n-1} ) | 11 |

131 | The 17 th term of an AP exceeds its 10 th term by ( 7 . ) Find the common difference A ( cdot frac{3}{7} ) B. ( frac{-3}{7} ) ( c cdot frac{7}{3} ) D. ( frac{-7}{3} ) | 11 |

132 | If ( 2,5, p ) are in ( A P ) find ( p ) A. 6 B. 8 c. 10 D. None of these | 11 |

133 | If ( x_{1}, x_{2}, dots, x_{n} ) are an observation such that ( sum_{i=1}^{n} x_{i}^{2}=400 ) and ( sum_{i=1}^{n} x_{i}=80 ),then the least value of ( n ) is A . 18 B. 12 ( c cdot 15 ) D. 16 | 11 |

134 | Write first four terms of the AP,when the first term ( a ) and the common difference ( d ) are given as follows: ( boldsymbol{a}=-mathbf{1 . 2 5}, boldsymbol{d}=-mathbf{0 . 2 5} ) | 11 |

135 | If ( a_{1}, a_{2}, dots a_{n} ) are positive real numbers whose product is a fixed number ( c ), then the minimum value of ( boldsymbol{a}_{1}+ ) ( boldsymbol{a}_{2}+, dots boldsymbol{a}_{boldsymbol{n}-1}+boldsymbol{2} boldsymbol{a}_{boldsymbol{n}} ) A ( cdot n(2 c)^{1 / n} ) B . ( (n+1) c^{1 / n} ) ( mathbf{c} cdot 2 n c^{1 / n} ) D. ( (n+1)(2 c)^{1 / n} ) | 11 |

136 | Solve: ( sum_{n=1}^{13}left(t^{n}-t^{n+1}right)= ) | 11 |

137 | If ( boldsymbol{A}=cos ^{2} boldsymbol{x}+frac{mathbf{1}}{cos ^{2} boldsymbol{x}}, boldsymbol{B}=cos boldsymbol{x}- ) ( frac{1}{cos x} forall x neq(2 n pm 1) frac{pi}{2}, ) then the minimum value of ( frac{boldsymbol{A}}{boldsymbol{B}} ) is A ( cdot sqrt{2} ) B. ( 2 sqrt{2} ) c. ( frac{1}{sqrt{2}} ) D. None of these | 11 |

138 | The sum of ( 2 n ) terms of a geometric progression whose first term is ‘ ( a^{prime} ) and common ratio ( ^{prime} r^{prime} ) is equal to the sum of ( n ) terms of a geometric progression whose first term is ‘b’ and common ‘ ( r^{2} ) ‘ then ( b ) is equal to A. The sum of the first two terms of the first series B. The sum of the first and last terms of the first series. c. The sum of the last two terms of the first series D. None of these | 11 |

139 | Let ( a_{n} ) be an A.P. for which ( a_{2}=20 ) and ( boldsymbol{a}_{mathbf{1 0}}=mathbf{4 0 .} ) Find ( boldsymbol{a}_{mathbf{5}} ) A. 0 B. 10 ( c cdot 20 ) D. 30 | 11 |

140 | Find the sum of the following infinite G.P.: ( frac{1}{3}, frac{-2}{9}, frac{4}{27}, frac{-8}{81}, dots ) A ( cdot frac{1}{2} ) B. ( c cdot frac{1}{3} ) D. | 11 |

141 | What is the ( 25^{t h} ) term of ( A P ) ( -5, frac{-5}{2}, 0, frac{5}{2} dots dots dots ) | 11 |

142 | If ( G . M .=18 ) and ( A . M .=27, ) then ( H . M . ) is A ( cdot frac{1}{18} ) B. ( frac{1}{12} ) ( c cdot 12 ) D. ( 9 sqrt{6} ) | 11 |

143 | In A.P. the first term is -4, the last term is 29 and the sum of all its terms is 150. Find its common difference. | 11 |

144 | The sum of the series ( sum_{r=1}^{n}(-1)^{r-1} cdot^{n} C_{r}(a-r) ) is ( A ) B. c. ( n .2^{n-1}+a ) D. None of these | 11 |

145 | Let ( A ) be the sum of the first 20 terms and ( B ) be the sum of the first 40 terms of the series ( 1+2.2^{2}+3^{2}+2.4^{2}+5^{2}+ ) ( 2.6^{2}+ldots . . . . . ) Find the value of ( boldsymbol{A} ) A . 496 в. 232 c. 248 D. 464 | 11 |

146 | ( 2.4+4.7+6.10+ldots(n-1) ) terms A ( cdot 2 n^{3}-2 n^{2} ) B. ( frac{n^{3}+3 n^{2}+1}{6} ) c. ( 2 n^{3}+2 n ) D. ( 2 n^{3}-n^{2} ) | 11 |

147 | Decide whether following sequence is an ( A . P ., ) if so find the ( 20^{t h} ) term of the progression ( -12,-5,2,9,16,236,30, dots . ) | 11 |

148 | What is the next term of the series ( 1+ ) ( 3+5+7+—? ) ( mathbf{A} cdot mathbf{9} ) B. 11 c. 10 D. 8 | 11 |

149 | If the 9 th term of an A.P is 35 and 19 th is ( 75, ) then its 20 th term will be ( mathbf{A} cdot 78 ) B. 79 c. 80 D. 81 | 11 |

150 | How many terms of an arithmetic progression must be taken for their sum to be equal to ( 91, ) if its third term is 9 and the difference between the seventh and the second term is ( 20 ? ) | 11 |

151 | 7. The sum of the series 1 1.1 ……… up to oo is equal to 1.2 2.3 3.4 (a) loge) (b) 2loge 2 (c) loge 2-1 (d) loge 2 | 11 |

152 | The A.M. of ‘n’ observations is M. If the sum of ( (boldsymbol{n}-mathbf{4}) ) observation is ‘a’, what is the mean of remaining 4 observations? A ( . n M+a ) в. ( frac{n M-a}{2} ) c. ( frac{n M+a}{2} ) D. ( frac{n M-a}{4} ) | 11 |

153 | The sum of the series ( frac{1}{(1 times 2)}+ ) ( frac{1}{(2 times 3)}+frac{1}{(3 times 4)}+ldots ldots+ ) ( frac{1}{(100 times 101)} ) is equal to A ( cdot frac{20}{1010} ) в. ( frac{100}{1010} ) c. ( frac{50}{101} ) D. ( frac{25}{101} ) | 11 |

154 | Use geometric series to express ( 0.555 ldots=0 . overline{5} ) as a rational number A ( cdot frac{1}{5} ) в. ( frac{5}{9} ) c. ( frac{5}{99} ) D. None of these | 11 |

155 | N, the set of natural numbers, is partitioned into subsets ( boldsymbol{S}_{1}= ) ( {1}, S_{2}={2,3}, S_{3}={4,5,6}, S_{4}= ) ( {7,8,9,10} . ) The last term of these groups is ( 1,1+2,1+2+3,1+2+ ) ( 3+4, ) so on. Find the sum of the elements in the subset ( S_{50} ) | 11 |

156 | Which of the following options is not a series? A. -2,-4,-6,-8,-10 0 в. -2,0,2,4,8 c. 1,2,3,4,5 D. All the above | 11 |

157 | Let a,,a,, ….. 0,be in A, P, and h, h….hobe in H.P. If a, = h, = 2 and a ,o=h10 = 3, then a h, is (1999 – 2 Marks) (2) 2 (6 3 1 (c) 5 (d) 6 | 11 |

158 | If the ( 14^{t h} ) term of an arithmetic series is 6 and ( 6^{t h} ) term is ( 14, ) then what is the ( 95^{t h} ) term? A . -75 B. 75 c. 80 D. – 80 | 11 |

159 | The ratio of the sum of first 3 terms to that of the first 6 terms of a ( G . P . ) is ( 125: 152 . ) Find their common ratio’s. | 11 |

160 | Assertion If ( a, b, c ) are three positive numbers in G.P., then ( left(frac{a+b+c}{3}right) cdotleft(frac{c+b}{a b+b c+c a}right)= ) ( (sqrt[3]{a b c})^{2} ) Reason (A.M.) (H.M.) ( =(G . M .)^{2} ) is true for any set of positive numbers. A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion C. Assertion is correct but Reason is incorrect D. Assertion is incorrect but Reason is correct | 11 |

161 | If ( frac{a^{n}+b^{n}}{a^{n-1}+b^{n-1}} ) is the A.M. between ( a ) and ( b, ) then find the value of ( n ) | 11 |

162 | Find the following sum. ( sum_{r=1}^{n}left(6 r^{2}-2 r+6right) ) | 11 |

163 | Find the sum of first 10 terms of the arithmetic series if ( a_{1}=2 ) and ( a_{10}=22 ) A. 155 в. 120 ( c .165 ) D. 130 | 11 |

164 | How many terms of the AP ( 20,19 frac{1}{3}, 18 frac{2}{3}, ldots ) must be taken to make the sum ( 300 ? ) Explain the double answer | 11 |

165 | Find the middle term of the AP ( 6,13,20, dots dots, .216 ) | 11 |

166 | The arithmetic mean of ( 1,8,27,64, dots ) up to n terms is given by A ( cdot frac{n(n+1)}{2} ) B. ( frac{n(n+1)^{2}}{2} ) c. ( frac{n(n+1)^{2}}{4} ) D. ( frac{n^{2}(n+1)^{2}}{4} ) | 11 |

167 | 2. For 0 < 0 5 18/2, if x = cos2n 0, y = sin?" 6, n=0 n=0 z= cos2n o sin210, then n=0 a. xyz = xz + y c. xyz = x + y + z b. xyz = xy + z d. xyz = yz + x | 11 |

168 | Find the common difference and write the next four terms of each of the following arithmetic progression: ( -1,-frac{5}{6},-frac{2}{3}, dots ) | 11 |

169 | If ( a, b ) and ( c ) are positive real numbers then ( frac{a}{b}+frac{b}{c}+frac{c}{a} ) is greater than or equal to. ( A cdot 3 ) B. 6 ( c cdot 27 ) D. 5 | 11 |

170 | Find the sum of the following A.P ( 2,7,12, dots . . ) to 10 terms | 11 |

171 | Find the common difference of the A.P. given below: ( mathbf{0 . 6}, mathbf{1 . 7}, mathbf{2 . 8}, mathbf{3 . 9}, ldots . ) | 11 |

172 | The sum of all odd proper divisors of 360 is A . 77 B. 78 ( c cdot 81 ) D. none of these | 11 |

173 | The sum of nn is equal to A ( cdot frac{1}{4} n(n+1)(n+2) ) B. ( frac{1}{4} n(n+1)(n+2)(n+3) ) c ( cdot frac{1}{2} n(n+1)(n+2)(n+3) ) D. None of these | 11 |

174 | Find the ( A . M . ) of the series ( mathbf{1}, mathbf{2}, mathbf{4}, mathbf{8}, mathbf{1 6}, dots dots mathbf{2}^{n} ) | 11 |

175 | Find the common ratio of the G.P. 2,-6 ( 18, dots dots ) ( A cdot-2 ) B . – – ( c cdot-4 ) D. – – | 11 |

176 | Sum of ( n ) terms of ( A ) is ( n^{2}+2 n . ) Find the first term and common difference | 11 |

177 | Which term of the sequence ( -1,3,7,11, dots ) is ( 95 ? ) | 11 |

178 | If ( a+b+c=1 ) and ( a, b, c ) are all distinct positive reals, then prove that (1- a) b) ( (1-c)>8 ) abc. | 11 |

179 | For what values of ( x, ) the numbers ( -frac{2}{7}, x,-frac{2}{7} ) are in ( G . P ? ) | 11 |

180 | Check if the sequence is an AP ( mathbf{1}, mathbf{3}, mathbf{9}, mathbf{2 7}, dots ) A. Yes, it is an AP B. No c. Data Insufficient D. Ambiguous | 11 |

181 | Let ( a, b, c ) are ( 7^{t h}, 11^{t h} ) and ( 13^{t h} ) terms of non constant AP. If ( a, b, c ) are also in GP, then find ( frac{boldsymbol{a}}{boldsymbol{c}} ) ( mathbf{A} cdot mathbf{1} ) B. 2 ( c cdot 3 ) D. 4 | 11 |

182 | The sum of the fourth and twelfth term of an arithmetic progression is 20 What is the sum of the first 15 terms of the arithmetic progression? A . 150 B. 200 ( c .250 ) D. 500 | 11 |

183 | The arithmetic mean of the following data is ( 25, ) find the value of ( k ) 35 ( begin{array}{llll}boldsymbol{x}_{boldsymbol{i}}: & boldsymbol{5} & boldsymbol{1} boldsymbol{5} & boldsymbol{2} boldsymbol{5}end{array} ) ( begin{array}{llll}f_{i:} & 3 & k & 3 & 6end{array} ) | 11 |

184 | Find the ( 12^{t h}, 24^{t h} ) and ( n t h ) term of the A.P. given by ( 9,13,17,21,25, dots ) | 11 |

185 | What is the common difference of an A.P., in which ( a_{21}-a_{7}=644 ) | 11 |

186 | The fire terms of an arithmetic sequence are given:- -7, -4, -1, 2, 5 ……… Find their common difference and next term of the sequence. A. common difference ( =3 ; a_{6}=8 ) B. Common difference ( =2 ; a_{4}=7 ) C. common difference ( =1 ; a_{6}=5 ) D. common difference ( =0 ; a_{6}=0 ) | 11 |

187 | ( operatorname{Let} a_{i}=i+frac{1}{t} ) for ( i=1,2, dots dots, 20 ) put ( boldsymbol{p}=frac{mathbf{1}}{mathbf{2 0}}left(boldsymbol{a}_{1}+boldsymbol{a}_{2}+ldots ldots+boldsymbol{a}_{20}right) ) and ( boldsymbol{q}= ) ( frac{1}{20}left(frac{1}{a_{1}}+frac{1}{a_{2}}+ldots . .+frac{1}{a_{20}}right) . ) Then A ( cdot q epsilonleft(0, frac{22-p}{21}right) ) В ( cdot_{q epsilon}left(frac{22-p}{21}, frac{2(22-p)}{21}right) ) c. ( _{q epsilon}left(frac{2(22-p)}{21}, frac{22-p}{7}right) ) D. ( q epsilonleft(frac{22-p}{7}, frac{4(22-p)}{21}right) ) | 11 |

188 | State whether the following sequence is an Arithmetic Progression or not. ( mathbf{3}, mathbf{6}, mathbf{1 2}, mathbf{2 4}, dots dots ) | 11 |

189 | If ( S_{n}=n P+frac{n(n-1) Q}{2}, ) where ( S n ) denotes the sum of the first ( n ) terms of an ( A P, ) then the common difference is ( mathbf{A} cdot P+Q ) в. ( 2 P+3 Q ) c. ( 2 Q ) D. ( Q ) | 11 |

190 | Find the sum of all two digit numbers which when divided by ( 4, ) yield unity as remainder. | 11 |

191 | If the sum of 8 terms of an A.P. is 64 and the sum of 19 terms is 361 , find the sum of ( n ) terms. | 11 |

192 | Is the following sequence an ( A P ? ) If true, find the common difference ( d ) and write three more terms. ( mathbf{0},-mathbf{4},-mathbf{8},-mathbf{1 2} ) | 11 |

193 | The sum to infinity of ( frac{1}{7}+frac{2}{7^{2}}+frac{1}{7^{3}}+ ) ( frac{2}{7^{4}}+dots ) is A ( cdot frac{1}{5} ) в. ( frac{7}{24} ) c. ( frac{5}{48} ) D. ( frac{3}{16} ) | 11 |

194 | If the slopes of the lines given by the equation ( 24 x^{3}-a x^{2} y+26 x y^{2}- ) ( 3 y^{3}=0 ) are in G.P., p is the greatest perpendicular distance of the point (1,1) from these lines, then ( a^{2}+37 p^{2} ) is equal to | 11 |

195 | If ( I_{n}=int_{0}^{pi / 4} tan ^{n} x d x, ) then ( frac{1}{I_{2}+I_{4}}, frac{1}{I_{3}+I_{5}}, frac{1}{I_{4}+I_{6}}, dots ) are in A. A.P B. G.P. c. н.P D. none | 11 |

196 | Identify the series. ( mathbf{A} cdot{1,2,3,4,5} ) B . ( 1+2+3+4+5 ) c. ( 1 times 2 times 3 times 4 ) D. ( 2-4 times 3+1-23 ) | 11 |

197 | If ( x+3 ) is the geometric mean between ( x ) +1 and ( x+6 ) then find ( x ) ( A cdot 2 ) B. 3 ( c cdot 4 ) D. | 11 |

198 | ( mathbf{1}^{3}+mathbf{1}^{2}+mathbf{1}+mathbf{2}^{3}+mathbf{2}^{2}+mathbf{2}+mathbf{3}^{3}+mathbf{3}^{2}+ ) ( mathbf{3}+ldots+mathbf{3 n} ) terms ( = ) ( mathbf{A} cdot frac{n(n+1)left(n^{2}+12 n+5right)}{12} ) | 11 |

199 | For an A.P., ( t_{3}=8 ) and ( t_{4}=12, ) find the common difference ( d ) | 11 |

200 | A fibonacci series is: A. series of numbers in which each number (Fibonacci number) is the sum of the two preceding numbers. B. the simplest is the series ( 1,1,2,3,5,8, ) etc. C. Both are correct D. None is correct | 11 |

201 | The geometric mean of 6 and 54 is A . 12 B. 16 c. 18 D . 20 | 11 |

202 | For the following arithmetic progressions write the first term ( a ) and the common difference ( d ) : ( -1.1,-3.1,-5.1,-7.1, dots . ) | 11 |

203 | The sum of first ( n ) terms of an ( A . P ) is ( 2 n+3 n^{2} . ) Find the common difference. | 11 |

204 | Observe the following patterns ( mathbf{1}=frac{mathbf{1}}{mathbf{2}}(mathbf{1} times(mathbf{1}+mathbf{1})) ) ( mathbf{1}+mathbf{2}=frac{mathbf{1}}{mathbf{2}}(mathbf{2} times(mathbf{2}+mathbf{1})) ) ( mathbf{1}+mathbf{2}+mathbf{3}=frac{mathbf{1}}{mathbf{2}}(mathbf{3} times(mathbf{3}+mathbf{1})) ) ( mathbf{1}+mathbf{2}+mathbf{3}+mathbf{4}=frac{mathbf{1}}{mathbf{2}}(mathbf{4} times(mathbf{4}+mathbf{1})) ) and find the value of each of the following [ mathbf{1}+mathbf{2}+mathbf{3}+mathbf{4}+mathbf{5}+ldots+mathbf{5 0} ] | 11 |

205 | If ( f(x)=frac{a^{x}}{a^{x}+sqrt{a}}(a>0), ) evaluate ( sum_{r=1}^{2 n-1} 2 fleft(frac{r}{2 n}right) cdot ) for ( n=8 ) | 11 |

206 | 10. Let a and B be the roots of x2-x-1=0, with a > B. For all positive integers n, define a-a”-B” a-Bn21 b = 1 and bn = an-1 + anti,n22 Then which of the following options is/are correct ? (JEE Adv. 2019) (a) jan – 10 10″ 89 (b) bn = an + Br for all n 2 1 © ay + a2 + az + ….. An = an+2 – 1 for all n 21 ob – 8 n=110″ 89 | 11 |

207 | Find the ( 9^{t h} ) term and the general term of the progression ( frac{1}{4},-frac{1}{2}, 1,-2, dots ) | 11 |

208 | The product of ( n ) positive integers is 1 then their sum is a positive integer, that is A. equal to 1 B. equal to ( n+n^{2} ) c. divisible by ( n ) D. never less than ( n ) | 11 |

209 | The mean of 3 observations is 12 and mean of 5 observations is 4 the combined mean is A. 7 B. 8 ( c cdot 9 ) D. 10 | 11 |

210 | If ( A M ) between ( p^{t h} ) and ( q^{t h} ) terms of an ( A P ) be equal to the AM between ( r^{t h} ) and ( s^{t h} ) term of the ( A P, ) then ( p+q ) is equal to A ( cdot r+s ) в. ( frac{r-s}{r+s} ) c. ( frac{r+s}{r-s} ) D. ( r+s+1 ) | 11 |

211 | If ( m ) th term of an AP is ( frac{1}{n} ) and ( n ) th is ( frac{1}{m} ) then show that ( (m n)^{t h} ) term of an AP is 1 | 11 |

212 | If the sum of first ( 2 n ) terms of ( A . P .2,5 ) ( 8, ldots ) is equal to the sum of the first ( n ) terms of the A.P.57,59, 61, …., then n equals- A . 10 B. 12 ( c cdot 11 ) D. 13 | 11 |

213 | Is the given sequence ( 3,3+sqrt{2}, 3+ ) ( 2 sqrt{2}, 3+3 sqrt{2} ) form an APs? If it forms an ( A P, ) find the common difference ( d ) and write the next three terms. | 11 |

214 | ( ln ) a series, ( boldsymbol{T}_{boldsymbol{n}}=boldsymbol{2} boldsymbol{n}+boldsymbol{5}, ) find ( boldsymbol{S}_{boldsymbol{4}} ) A .40 B. 30 c. 20 D. 10 | 11 |

215 | The first two terms of an A.P. are 27 and 24 respectively. How many terms of the progression are to be added to get ( -30 ? ) A . 15 B. 20 c. 25 D. 18 | 11 |

216 | Sum of certain consecutive odd positive integers is ( 57^{2}-13^{2} . ) Find them | 11 |

217 | The sum of first three terms of a ( G . P ) is to the sum of the first six terms as ( 125: 152 . ) Find the common ratio of the ( G . P ) A ( cdot frac{3}{5} ) в. ( frac{5}{3} ) ( c cdot frac{2}{3} ) D. ( frac{3}{2} ) | 11 |

218 | ( sum_{r=1}^{n} r(n-r) ) ( mathbf{A} cdot frac{1}{6} n(n+1)(2 n+1) ) B ( cdotleft(frac{n(n+1)}{2}right)^{2} ) C ( cdot frac{n^{2}(n+1)}{6} ) D. ( frac{nleft(n^{2}-1right)}{6} ) | 11 |

219 | If the roots of the equation ( x^{3}-a x^{2}+ ) ( 4 x-8=0 ) are real and positive, then the minimum value of ( a ) is | 11 |

220 | Find the sum to infinite terms of the series ( frac{7}{5}left(1+frac{1}{10^{2}}+frac{1.3}{1.2} cdot frac{1}{10^{4}}+frac{1.3 .5}{1.2 .3} cdot frac{1}{10^{6}}+right. ) | 11 |

221 | Sum to infinite terms the following series: ( 1+4 x+7 x^{2}+10 x^{3}+ldots,|x|<1 ) | 11 |

222 | State the following statement is True or False Arithmetic mean of first five natural numbers is 3 A . True B. False | 11 |

223 | For an A.P. given below find ( t_{20} ) and ( S_{10} ) ( frac{1}{6}, frac{1}{4}, frac{1}{3}, dots ) A ( cdot frac{7}{4}, frac{65}{12} ) в. ( frac{5}{4}, frac{63}{12} ) c. ( frac{5}{4}, frac{65}{12} ) D. ( frac{7}{4}, frac{63}{12} ) | 11 |

224 | If the sum of first ( p ) terms of an A.P.is equal to the sum of the first ( q ) terms, then find the sum of the first ( (boldsymbol{p}+boldsymbol{q}) ) terms. | 11 |

225 | 10,20,40,80 is an example of A . fibonacci sequence B. harmonic sequence C. arithmetic sequence D. geometric sequence | 11 |

226 | Write the first five terms of the following arthmetic progression where first term ( a=3, ) common difference ( d=4 ) | 11 |

227 | The sum of first three terms of a geometric sequence is ( frac{13}{12} ) and their product is ( -1 . ) Find the common ratio and the terms | 11 |

228 | If ( n ) geometric means be inserted between ( a ) and ( b ), then prove that their products is ( (boldsymbol{a} boldsymbol{b})^{boldsymbol{n} / 2} ) | 11 |

229 | The sum of the first n terms of an A.P. is ( 3 n^{2}+6 n . ) Find the nth term of this A.P. | 11 |

230 | In the following table, given that ( a ) is the first term, ( d ) the common difference and ( a_{n} ) the ( n t h ) term of the ( A P ) (i) (ii) ( begin{array}{ccccc}a & d & n & a_{n} \ 7 & 3 & 8 & dots \ -18 & dots & 10 & 0 \ dots & -3 & 18 & -5 \ -18.9 & 2.5 & dots & 3.6 \ 3.5 & 0 & 105 & dotsend{array} ) (iii) (iv) ( (v) ) A. ( (i) a_{n}=13(i i) d=1(i i i) a=46(i v) n=9(v) a_{n}=3.5 ) B. ( (i) a_{n}=36(i i) d=7(i i i) a=46(i v) n=13(v) a_{n}=3.5 ) C ( cdot(i) a_{n}=18(i i) d=3(i i i) a=46(i v) n=4(v) a_{n}=3.5 ) D ( cdot(i) a_{n}=28(i i) d=2(i i i) a=46(i v) n=10(v) a_{n}=3.5 ) | 11 |

231 | Devendra invested in a national saving certificate scheme. In the first year he invested Rs.1000 in second year Rs. ( 1400, ) in the third year ( R s .1800 ) and ( s 0 ) on. Find the total amount that he invested in 12 years | 11 |

232 | 32. For any three positive real numbers a, b and c, 9(25a² + b2) + 25(c2 – 3ac) = 15b(3a + c). Then : [JEE M2017] (a) (c) a, b and care in G.P. b, c and a are in A.P. (b) b c and a are in G.P. (d) a, b and c are in A.P. | 11 |

233 | For the following AP, write the first term and the common difference -5,-1,3,7 A. First term: 7 and Common difference: 7 B. First term: 3 and Common difference: -1 C. First term: -5 and Common difference: 4 D. First term: 3 and Common difference: -2 | 11 |

234 | If ( frac{a^{n}+b^{n}}{a^{n-1}+b^{n-1}} ) is the AM between a and b, then the value of ( n ) is A. B. ( c cdot-1 ) D. none of these | 11 |

235 | Insert 17 arithmetic means between ( 3 frac{1}{2} ) and ( -41 frac{1}{2} ) | 11 |

236 | ( ln ) an ( A P, operatorname{given} a=2, d=8, S_{n}=90 ) find ( n ) and ( a_{n} ) | 11 |

237 | Which of the following is a general form of geometric sequence? ( mathbf{A} cdot{2,4,6,8,10} ) B . {-1,2,4,8,-2} | 11 |

238 | Check whether 7,49,343 are in continued proportion or not | 11 |

239 | The 4 th term of a G.P. is square of its second term, and the first term, and the first term is ( -3 . ) Find its ( mathbf{7}^{t h} ) term. | 11 |

240 | Find ( a_{20} ) of a geometric sequence if the first few terms of the sequence are ( operatorname{given} ) by ( -frac{1}{2}, frac{1}{4},-frac{1}{8}, frac{1}{16}, dots ) | 11 |

241 | 15 2 a (21-1). > a; and 1 = 30 Let a,,a,, ….., a,, be an A.P., S= – i=1 If a = 27 and S-2T = 75, then a,, is equal to: JEEM 2019-9 Jan (M) (a) 52 (b) (c) 47 (d) 20 T… | 11 |

242 | ( frac{a^{n}+b^{n}}{a^{n-1}+b^{n-1}} ) is the AM between ( a ) and ( b ) if ( n ) is ( mathbf{A} cdot mathbf{0} ) в. ( frac{1}{2} ) ( c cdot 1 ) D. – | 11 |

243 | What is the next term to this series, 2 ( 3,7,16,32, ) and ( 57, dots ) A . 94 B. 93 c. 92 D. 95 | 11 |

244 | Sum the series: ( mathbf{2}^{1 / 4} cdot mathbf{4}^{mathbf{1} / 8} cdot mathbf{8}^{mathbf{1} / mathbf{1 6}} cdot mathbf{1 6}^{mathbf{1} / 32} ldots . ) is equal | 11 |

245 | If the 5 th term of an A.P is eight times the first and 8 th term exceeds twice the 4 th term by 3 , then the common difference is ( A cdot 7 ) B. 5 ( c .6 ) ( D ) | 11 |

246 | Find the mean of the first 10 natural numbers. | 11 |

247 | If sum of ( n ) terms of a sequence is given by ( S_{n}=2 n^{2}+3 n, ) find its ( 50^{t h} ) term. A . 250 в. 225 ( c cdot 201 ) D. 205 | 11 |

248 | If a and b are two unequal positive numbers, the: A ( cdot frac{2 a b}{a+b}>sqrt{a b}>frac{a+b}{2} ) B . ( sqrt{a b}>frac{2 a b}{a+b}>frac{a+b}{2} ) C ( cdot frac{2 a b}{a+b}>frac{a+b}{2}>sqrt{a b} ) D ( cdot frac{a+b}{2}>frac{2 a b}{a+b}>sqrt{a b} ) E ( cdot frac{a+b}{2}>sqrt{a b}>frac{2 a b}{a+b} ) | 11 |

249 | The sum of 15 terms of an arithmetic progression is 600 , and the common difference is ( 5, ) then the first term is A . 3 B. 4 ( c .5 ) D. none of these | 11 |

250 | State True or False. ( 1+frac{1}{5}+frac{3}{5^{2}}+frac{5}{5^{3}}+ldots . . infty=frac{13}{8} ) A . True B. False | 11 |

251 | ( ln operatorname{an} A P, operatorname{given} a=5, d=3, a_{n}=50 ) find ( n ) and ( S_{n} ) | 11 |

252 | Find the next term of the A.P. ( sqrt{8}, sqrt{18}, sqrt{32} dots dots ) | 11 |

253 | The arithmetic mean of ( 1,2,3, dots, n, ) is A ( cdot frac{n-1}{2} ) в. ( frac{n+1}{2} ) c. ( frac{n}{2} ) D. ( frac{n}{2}+1 ) | 11 |

254 | The mean of the cubes of the first ( n ) natural numbers is : A ( cdot frac{n(n+1)^{2}}{4} ) B ( cdot n^{2} ) c. ( frac{n(n+1)(n+2)}{8} ) D. ( left(n^{2}+n+1right) ) | 11 |

255 | Find the common difference ( d ) and write three more terms.2,4,8,16, | 11 |

256 | The arithmatic mean of 4,6,8 is ( A cdot 4 ) B. 6 ( c .8 ) D. 4.5 | 11 |

257 | The sum of series ( frac{1^{2}}{1}+frac{1^{2}+2^{2}}{1+2}+ ) ( frac{1^{2}+2^{2}+3^{2}}{1+2+3}+ldots . ) upto ( n ) terms is A ( cdot frac{1}{3}(2 n+1) ) B. ( frac{1}{3} n^{2} ) c. ( frac{1}{3}(n+2) ) D. ( frac{1}{3} n(n+2) ) | 11 |

258 | Find the sum of the products of the corresponding terms of the sequences 2,4,8,16,32 and ( 128,32,8,2, frac{1}{2} ) | 11 |

259 | Find the sum of ( n ) terms of ( 1^{2}+ ) ( left(1^{2}+2^{2}right)+left(1^{2}+2^{2}+3^{2}right)+ ) ( left(1^{2}+2^{2}+3^{2}+4^{2}right)+ldots ) from that find the sum of the first 10 terms | 11 |

260 | The sum of the following series ( 1+6+ ) ( frac{mathbf{9}left(mathbf{1}^{2}+mathbf{2}^{2}+mathbf{3}^{2}right)}{mathbf{7}}+ ) ( frac{12left(1^{2}+2^{2}+3^{2}+4^{2}right)}{9}+ ) ( frac{15left(1^{2}+2^{2}+ldots+5^{2}right)}{11}+ldots . ) up to 15 terms is: A . 7820 в. 7830 c. 7520 D. 7510 | 11 |

261 | Which one of the following statements is correct? ( mathbf{A} cdot mathbf{G}_{1}>mathbf{G}_{2}>mathbf{G}_{3}> ) B ( cdot mathrm{G}_{1}<mathrm{G}_{2}<mathrm{G}_{3}<ldots ) c. ( mathrm{G}_{1}=mathrm{G}_{2}=mathrm{G}_{3}= ) D. ( G_{1}<G_{3}<G_{5}G_{4}>G_{6}>ldots ) | 11 |

262 | Find the ( n^{t h} ) term of the Geometric Progression. ( mathbf{1 0 0},-mathbf{1 1 0}, mathbf{1 2 1}, dots ) | 11 |

263 | Find the number of terms of the AP: ( -12,-9,-6, dots 12 . ) If 1 is added to each term of this ( A P, ) then find the sum of all terms of the AP thus obtained. | 11 |

264 | If ( p, q, r ) are positive and are in A.P., then the roots of the quadratic equation ( p x^{2}+q x+r=0 ) are real for A ( cdotleft|frac{r}{p}-7right| geq 4 sqrt{3} ) В ( cdotleft|frac{p}{r}-7right|<4 sqrt{3} ) c. all ( p ) and ( r ) D. no ( p ) and ( r ) | 11 |

265 | f ( p, p+2, p+6 ) are in GP find ( p ) | 11 |

266 | If for an ( A, P ., S_{8}=16 ) and ( S_{16}=8, ) find the first negative term | 11 |

267 | The sum of 100 terms of the series. ( 9+ ) ( mathbf{0 9}+.009 . ) will be? ( ^{mathbf{A}} cdot_{1}-left(frac{1}{10}right)^{100} ) B. ( 1+left(frac{1}{10}right)^{106} ) ( ^{mathrm{c}} cdot_{1-}left(frac{1}{10}right)^{106} ) D. ( _{1+}left(frac{1}{10}right)^{100} ) | 11 |

268 | Adding and constant difference between the terms is called A. sequence B. constant c. term D. series | 11 |

269 | The ( 4^{t h} ) term of A.P is equal to 3 times the first term and the ( 7^{t h} ) term excess which the third term by 1. Find its ( n^{t h} ) term. ( mathbf{A} cdot n+2 ) B. ( 3 n+1 ) c. ( (2 n+1) ) D. ( 3 n+2 ) | 11 |

270 | On Monday morning Mr. Smith had a certain amount of money that he planned to spend during the week. On each subsequent morning, he had one fourth the amount of the previous morning. On Saturday morning, 5 days later, he had ( $ 1 . ) How many dollars did Mr. Smith originally start with on Monday morning?(Disregard the ( $ operatorname{sign} ) when gridding your answer. | 11 |

271 | If an AP, the sum of first ten term is -150 and the sum of its next ten terms is -550. Find first term of AP. | 11 |

272 | Write the first five terms of the sequence and obtain the corresponding series: ( boldsymbol{a}_{1}=boldsymbol{a}_{2}=boldsymbol{2}, boldsymbol{a}_{n}=boldsymbol{a}_{n-1}-mathbf{1}, boldsymbol{n}>mathbf{2} ) | 11 |

273 | If they are A.P. find the common difference, 2,4,6,8 | 11 |

274 | If ( A ) and ( G ) be ( A . M ) and ( G . M, ) respectively between two positive numbers. prove that the numbers are ( boldsymbol{A} pm ) ( sqrt{(boldsymbol{A}+boldsymbol{G})(boldsymbol{A}-boldsymbol{G})} ) | 11 |

275 | If ( a_{1}, a_{2}, dots dots, a_{24} ) are in ( A P ) and ( a_{1}+ ) ( boldsymbol{a}_{boldsymbol{5}}+boldsymbol{a}_{boldsymbol{1 0}}+boldsymbol{a}_{boldsymbol{1 5}}+boldsymbol{a}_{boldsymbol{2 0}}+boldsymbol{a}_{boldsymbol{2 4}}=mathbf{2 2 5} ) then the sum of 24 terms of this AP is. A. 900 в. 450 ( c cdot 225 ) D. None of these | 11 |

276 | If for a sequence ( left(boldsymbol{t}_{n}right), boldsymbol{S}_{n}=mathbf{4} boldsymbol{n}^{2}-boldsymbol{3} boldsymbol{n} ) show that the sequence is an ( A . P ) | 11 |

277 | Let ( T_{r} ) be the rth term of an A.P. whose first term is ( a ), and common difference is ( d ). If for some positive integers ( boldsymbol{m}, boldsymbol{n}, boldsymbol{T}_{boldsymbol{m}}=frac{mathbf{1}}{boldsymbol{n}} ) and ( boldsymbol{T}_{boldsymbol{n}}=frac{mathbf{1}}{boldsymbol{m}} ) then ( boldsymbol{a}-boldsymbol{d}= ) ( mathbf{A} cdot mathbf{0} ) B. ( frac{1}{m}+frac{1}{n} ) c. ( frac{1}{m n} ) D. | 11 |

278 | 27. The sum of first 9 terms of the series. 1 1 +23 1 +2 +33 1 1+ 3 1 +3+5 a) 142 (b) 192 -t – +- – +…. [JEE M 2015] (d) 96 (c) 71 | 11 |

279 | The first term of an AP is 5, last term is 45 and the sum is ( 400 . ) Find the number of terms and the common difference. A. ( n=16 ) and ( d=8 / 3 ) B. ( n=16 ) and ( d=16 / 3 ) c. ( n=8 ) and ( d=16 / 3 ) D. ( n=8 ) and ( d=8 / 3 ) | 11 |

280 | Divide 28 into four parts in A.P. so that ratio of the product of first and third with the product of second and fourth is 8:15.Now find the largest term in those four terms ? | 11 |

281 | The sum of first three terms of a G.P. is ( mathbf{3 9} ) ( frac{w}{10} ) and their product is ( 1 . ) Find the common ratio and the terms. | 11 |

282 | Find the sum of the following arithmetic progressions: ( 1,3,5,7, dots . ) to 12 terms | 11 |

283 | Directions : Find the missing number from the given responses. 21. 1 24 4 59 3 2 50 70 ? (1) 23 (2) 115 (3) 118 (4) 220 | 11 |

284 | ( ln ) an ( A . P 17^{t h} ) term is 7 more then is ( 10^{t h} ) term. Find the common difference? | 11 |

285 | The seventh term of a G.P. is 8 times the fourth term and 5th term is ( 48 . ) Find the second term of G.P. | 11 |

286 | How many terms of the series ( 2+6+ ) ( 18+. ) must be taken to make the sum equal to ( 728 ? ) | 11 |

287 | The least area of a circle circumscribing any right triangle of area S is: A . ( pi S ) в. ( 2 pi S ) c. ( sqrt{2} pi S ) D. ( 4 pi S ) | 11 |

288 | The value of ( 1+2+4+8 ldots . ) of G.P. where ( n=6 ) is A . 61 B. 62 ( c cdot 63 ) D. 64 | 11 |

289 | If ( a^{2}, b^{2}, c^{2} ) are in A.P., then the following are also in A.P. ( frac{boldsymbol{a}}{boldsymbol{b}+boldsymbol{c}}, frac{boldsymbol{b}}{boldsymbol{c}+boldsymbol{a}}, frac{boldsymbol{c}}{boldsymbol{a}+boldsymbol{b}} ) A. True B. False | 11 |

290 | The Sum of three numbers in AP is 75 and product of extremities is ( 609 . ) The numbrs and AM of 1st two numbers is ( mathbf{A} cdot{21,25,29}, ) АМ ( =23 ) B. ( {13,17,21}, ) АМ ( =22 ) c. ( {21,25,29}, ) АМ ( =25 ) D. ( {21,22,29}, ) АМ ( =23 ) | 11 |

291 | The sum of all the integers from 1 to 100 which are divisible by 2 or 5 is A. 3000 B. 3050 ( c .3600 ) D. 3250 | 11 |

292 | Find the mean of 43,54,64,53,36 A . 50 B. 40 ( c .60 ) D. 30 | 11 |

293 | What is the common difference of an AP whose nth term is ( boldsymbol{x} boldsymbol{n}+boldsymbol{y} ? ) ( mathbf{A} cdot x+y ) B. ( y ) ( c ) D. – ( x ) | 11 |

294 | Find the ( 19^{t h} ) term of the following A.P. ( mathbf{7}, mathbf{1 3}, mathbf{1 9}, mathbf{2 5}, dots ) | 11 |

295 | ( ln ) a certain G.P. if ( S_{6}=126 ) and ( S_{3}=14 ) then find ( a ) and ( r ) | 11 |

296 | ( ln operatorname{an} A P t_{1}+t_{7}=18, t_{4}+t_{10}=54 ) find twenty value of 520 | 11 |

297 | There are 60 terms in an A.P. of which the first term is 8 and the last term is 185. The ( 31^{s t} ) term is A . 56 B. 94 c. 85 D. 98 | 11 |

298 | In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of this progression equals A ( cdot frac{1}{2}(1-sqrt{5}) ) B ( cdot frac{1}{2} sqrt{5} ) c. ( frac{1}{5} sqrt{2} ) D. ( frac{1}{2}(sqrt{5}-1) ) | 11 |

299 | Is the given sequence 2,4,8,16 form an A.P. If it forms an AP, find the common difference ( d ) | 11 |

300 | The G.M. between 1 and 25 is | 11 |

301 | Find the sum of: 1. The first 50 multiples of 11 2. The multiples of 7 between 0 and ( mathbf{1 0 0 0} ) 3. The integers between 1 and 100 which are not divisible by 3 and 7 | 11 |

302 | Sum of 4 numbers in GP is ( 60 . ) And the AM of first and last no. is 18 find the first term and common difference of the ( mathrm{GP} ) This question has multiple correct options A ( . a=4, r=2 ) B. ( a=32, r=frac{1}{2} ) c. ( a=3, r=1 ) D. ( a=6, r=3 ) | 11 |

303 | Find the next term ( mathbf{3}, mathbf{5}, mathbf{7}, mathbf{1 0}, mathbf{1 1}, mathbf{1 5}, mathbf{1 5}, mathbf{2 0}_{mathbf{-}} ) | 11 |

304 | Which of the following list of numbers does form an AP? A. ( 2,4,8,16, dots ) В ( cdot 2, frac{5}{2}, 3, frac{7}{2}, ldots ) c. ( 0.2,0.22,0.222,0.2222, ldots ) D. ( 1,3,9,27, ldots ) | 11 |

305 | In an AP the sum of first n terms is ( frac{3 n^{2}}{2}+frac{5 n}{2} ) Find ( 25^{t h} ) term | 11 |

306 | A.M. of ( a-2, a, a+2 ) is | 11 |

307 | Find the seventh term of the geometric sequence ( 1,-3,9,-27, dots ) A . – 243 B. -30 ( c cdot 81 ) D. 189 E . 729 | 11 |

308 | Series can be defined as: A. a number of things or events that are arranged or happen one after the other. B. a set of regularly presented television shows involving the same group of characters or the same subject. C. set of books, articles, etc., that involve the same group of characters or the same subject D. All of the above | 11 |

309 | The third term of a geometric progression is 4. The product of the first five terms is (1982 – 2 Marks) (a) 43 (b) 45 (c) 44 (d) none of these | 11 |

310 | Find the sum of ( 7+10 frac{1}{2}+14+ldots+84 ) A ( cdot frac{2093}{2} ) в. ( frac{1093}{2} ) c. ( frac{2193}{2} ) D. ( frac{3093}{2} ) | 11 |

311 | Find the sum of the first 50 even positive integers. | 11 |

312 | Calculate the missing term in the geometric series ( frac{1}{4}+x+frac{1}{36}+frac{1}{108}+ ) A в. c. ( frac{1}{12} ) D. ( frac{1}{16} ) E ( frac{1}{18} ) | 11 |

313 | 6. If sin 0, tan , cos are in G.P. then 4 sin 0 – 3 sino + sinº0= | 11 |

314 | Find the sum of the series ( 1.3^{2}+ ) ( 2.5^{2}+3.7^{2}+ ) to ( n ) terms. | 11 |

315 | A right triangle is drawn in a semicircle of radius ( frac{1}{2} ) with one of its legs along the diameter. The maximum area of the triangle is A ( cdot frac{1}{4} ) B. ( frac{3 sqrt{3}}{32} ) ( c cdot frac{3 sqrt{3}}{16} ) D. | 11 |

316 | What is the common difference, if ( a_{1}= ) 100 and ( a_{2}=250 ) A. 100 в. 150 c. 200 D. 210 | 11 |

317 | If the ( 2^{n d} ) term of an AP is 13 and the ( 5^{t h} ) term is 25 what is its ( 7^{t h} ) term? | 11 |

318 | Find the next term of the sequence: ( mathbf{3}, mathbf{6}, mathbf{1 2}, mathbf{2 4}, dots dots dots ) A . 32 B. 48 c. 54 D. 64 | 11 |

319 | Which term of the A.P. ( mathbf{2 1}, mathbf{4 2}, mathbf{6 3}, mathbf{8 4}, dots dots dots mathbf{~ i s} mathbf{2 1 0} ) ( mathbf{A} cdot 9^{t h} ) B. ( 10^{text {th }} ) c. ( 11^{text {th }} ) D. ( 12^{text {th }} ) | 11 |

320 | For three number ( a, b, c ) product of the average of the number ( a^{2}, b^{2}, c^{2} ) and ( frac{1}{a^{2}} ) ( frac{1}{b^{2}}, frac{1}{c^{2}} ) can not be less than ( A ) B. 3 ( c cdot 9 ) D. none of these | 11 |

321 | The number of the terms of a geometric progression is even. The sum of all terms of the progression is thrice as large as the sum of its odd terms. Find the common ratio of the progression. | 11 |

322 | Complete the following table | 11 |

323 | Write down next three numbers to continue the pattern. 1,76,( 300 ; 1,76,400 ) | 11 |

324 | The sum of 100 terms of the progression ( 5,5,5, dots dots ) | 11 |

325 | ( ln operatorname{an} A . P, ) if ( m^{t h} ) term is ( n ) and ( n^{t h} ) term is ( mathrm{m} . ) Then find ( p^{t h} ) term ( (boldsymbol{m} neq boldsymbol{n}) ) | 11 |

326 | Let ( a_{1}, a_{2}, dots ) be positive real numbers in geometric progression. For each ( n ), let ( A_{n}, G_{n}, H_{a} ) be respectively, the arithmetic mean, geometric mean and harmonic mean of ( a_{1}, a_{2}, dots, a_{n} . ) Find an expression for the geometric mean of ( G_{1}, G_{2}, ldots, G_{n} ) in terms of ( boldsymbol{A}_{1}, boldsymbol{A}_{2}, ldots ., boldsymbol{A}_{n}, boldsymbol{H}_{1}, boldsymbol{H}_{2}, ldots . ., boldsymbol{H}_{n} ) | 11 |

327 | The first term of an A.P. is 5 and its common difference is ( -3 . ) Find the ( 11^{t h} ) term of an A.P. | 11 |

328 | The A.P. in which ( 8^{t h} ) term is -15 and ( 9^{t h} ) term is ( -30 . ) Find the sum of the first 10 numbers. | 11 |

329 | If ( a^{2}, b^{2}, c^{2} ) are in A.P. then show that ( frac{1}{b+c}, frac{1}{c+a}, frac{1}{a+b} ) are also in A.P. | 11 |

330 | 22. If Sn = cot'(3) + cot-‘(7) + cot-‘(13) + cot ‘(21) +… n terms, then a. So = tan 1 37 b. So – 17 – 2 c. So = sin 7 d. 20 = cot 1.1 | 11 |

331 | Decide wheather following sequence is an ( A . P ) if so find the ( 20^{circ} ) term of the progression. ( -12,-5,2,9,16,23,30, dots . ) | 11 |

332 | For three numbers ( a, b, c ) product of the average of the numbers, ( a^{2}, b^{2}, c^{2} ) and ( frac{1}{a^{2}}, frac{1}{b^{2}}, frac{1}{c^{2}} ) cannot be less than A . 1 B. 3 ( c .9 ) D. none of these | 11 |

333 | If ( a, b, c ) are distinct and the roots of ( (b- ) c) ( x^{2}+(c-a) x+(a-b)=0 ) are equal ,then ( a, b, c ) are in A. Arithmetic progression B. Geometric progression c. Harmonic progression D. Arithmetico-Geometric progression | 11 |

334 | If ( A_{1} A_{2} ; G_{1} G_{2} ; H_{1} H_{2} ) be two A.M.s G.M.s and H.M.s between two numbers then prove: ( frac{boldsymbol{G}_{1} boldsymbol{G}_{2}}{boldsymbol{H}_{1} boldsymbol{H}_{2}}=frac{boldsymbol{A}_{1}+boldsymbol{A}_{2}}{boldsymbol{H}_{1}+boldsymbol{H}_{2}} ) | 11 |

335 | In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of its progression is equals [2007] (a) 15 (b) (15-1) (2) 1 5. (0) 1 (1-15) un to 1 | 11 |

336 | Find the sum up to 30 terms of an ( A P ) whose second term is ( frac{1}{2} ) and ( 29^{t h} ) term is ( 49 frac{1}{2} ) | 11 |

337 | If ( frac{a^{n}+b^{n}}{a^{n-1}+b^{n-1}} ) is the G.M between ( a ) and ( b ) then the value of ( n ) is A. 0 B. ( c cdot frac{1}{2} ) D. | 11 |

338 | Find the value of ( x ) such that ( -frac{2}{7}, x,-frac{7}{2} ) are three consecutive terms of a G.P. | 11 |

339 | Adding first 100 terms in a sequence is called A. term B. series c. constant D. sequence | 11 |

340 | Find three consecutive terms in A.P. such that their sum is 21 and their products is 315 | 11 |

341 | If the ( 4^{t h} ) and ( 7^{t h} ) terms of a G.P. are 54 and 1458 respectively, find the G.P. | 11 |

342 | Find the terms ( a_{2}, a_{3}, a_{4} ) and ( a_{5} ) of a geometric sequence if ( a_{1}=10 ) and the common ratio ( r=-1 ) | 11 |

343 | If four numbers are in A.P. such that their sum is 60 and the greatest number is 4 times the least, then the numbers are A . 5,10,15.20 B. 4,10,16,22 c. 3,7,11,15 D. None of these | 11 |

344 | Find the Odd one among : 123,14,246 56,369,125 A . 123 B. 14 c. 246 D. 125 | 11 |

345 | If ( a, b, c ) are the sides of a triangle and ( s ) ( =frac{a+b+c}{2}, ) then prove that ( 8(s-a)(s- ) b) ( (s-c) leq a b c ) | 11 |

346 | A sum of an infinite sequence it is called a A. term B. constant c. series D. sequence | 11 |

347 | The cubes of the natural numbers are ( operatorname{grouped~as} 1^{3},left(2^{3}, 3^{3}right),left(4^{3}, 5^{3}, 6^{3}right) dots ) then sum of the numbers in the nth group is A ( cdot frac{n^{2}}{12}left(n^{2}+1right)left(n^{2}+4right) ) B ( frac{n^{3}}{8}left(n^{2}+1right)left(n^{2}+3right) ) c. ( frac{n^{3}}{8}left(n^{2}+1right)left(n^{2}+4right) ) D ( frac{n^{2}}{16}left(n^{2}+1right)left(n^{2}+4right) ) | 11 |

348 | If ( a+b+ldots+l ) are in ( G . P ., ) then prove that its sum ( frac{b l-a^{2}}{b-c} ) | 11 |

349 | In an A.P., if the first term is 22 , the common difference is -4 and the sum to ( n ) terms is ( 64, ) find ( n ) | 11 |

350 | The minimum value of ( 4^{x}+4^{1-x}, x epsilon R ) is ( A cdot 2 ) B. 4 ( c ) D. none of these | 11 |

351 | If the sum of ( p ) terms of an A.P. is q and the sum of q terms is ( p ), then the sum of ( p+q ) terms is ( A cdot O ) B. p-q ( c cdot p+1 ) ( D cdot-(p+q) ) | 11 |

352 | If ( a, b ) and ( c ) are three positive real numbers, then the minimum value of the expression ( frac{b+c}{a}+frac{c+a}{b}+frac{a+b}{c} ) is: A . B . 2 ( c cdot 3 ) D. | 11 |

353 | If ( boldsymbol{S}=mathbf{1}+frac{mathbf{1}}{mathbf{2}}+frac{mathbf{1}}{mathbf{4}}+frac{mathbf{1}}{mathbf{8}}+frac{mathbf{1}}{mathbf{1 6}}+frac{mathbf{1}}{mathbf{3 2}}+ ) ( dots infty ) then, the sum of the given series is 2 A. True B. False | 11 |

354 | If ( a_{1}, a_{2}, a_{3}, dots, a_{n} ) are positive real numbers whose product is a fixed number ( c ), then the minimum value of ( boldsymbol{a}_{1}+boldsymbol{a}_{2}+boldsymbol{a}_{3}+ldots ldots+boldsymbol{a}_{n-1}+boldsymbol{a}_{n} ) is A ( cdot n(c)^{1 / n} ) B cdot ( (n+1) c^{1 / n} ) ( mathbf{c} cdot 2 n c^{1 / n} ) D cdot ( (n+1)(2 c)^{1 / n} ) | 11 |

355 | The twentieth term of the GP ( frac{5}{2}, frac{5}{4}, frac{5}{8}, dots ) is A ( cdot frac{5}{2^{19}} ) в. ( frac{5}{2^{20}} ) c. ( frac{5}{2^{10}} ) D. ( frac{5}{10^{11}} ) | 11 |

356 | If you have a finite geometric sequence, the first number is 2 and the common ratio is ( 4, ) what is the 3 rd number in the sequence? ( A cdot 8 ) B. 16 ( c cdot 32 ) D. 64 | 11 |

357 | Find three consecutive odd numbers whose sum is 147 | 11 |

358 | The sum of ( 5^{t h} ) and ( 9^{t h} ) terms of an A.P. is 30. If its ( 25^{t h} ) term is three times its ( 8^{t h} ) term, find the AP. | 11 |

359 | The first term of an ( A . P . ) is ( 5, ) the common difference is 3 and the last term is ( 80, ) find the number of terms. | 11 |

360 | Let ( S ) be the infinite sum given by ( S= ) ( sum_{n=0}^{infty} frac{a_{n}}{10^{2 n}}, ) where ( left(a_{n}right)_{n geq 0} ) is a sequence defined by ( a_{0}=a_{1}=1 ) and ( a_{j}= ) ( 20 a_{j-1} ) for ( j geq 2 . ) If ( S ) is expressed in the form ( frac{a}{b}, ) where ( a, b ) are coprime positive integers, than ( a ) equals. A . 60 B. 75 c. 80 D. 81 | 11 |

361 | UUDU 9. Ifin a AABC, the altitudes from the vertices A, B, Con opposite sides are in H.P, then sin A, sin B, sin Care in [2005] (a) GP. (b) A.P. (c) A.P-G.P. (d) H.P | 11 |

362 | ( ln operatorname{an} A . P . ) if ( frac{S_{m}}{S_{n}}=frac{m^{4}}{n^{4}} ) then prove that ( frac{T_{m+1}}{T_{n+1}}=frac{(2 m+1)^{3}}{(2 n+1)^{3}} ) | 11 |

363 | 35. Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series 12 +2:22 +32 +2.42 +52 + 2.62 +… If B-2A = 1002, then 2 is equal to : [JEE M 2018 (a) 248 (b) 464 (c) 496 (d) 232 | 11 |

364 | An AP has the property that the sum of first ten terms is half the sum of next ten terms. If the second term is ( 13, ) then the common difference is A . 3 B. 2 c. 5 D. 4 E . 6 | 11 |

365 | Consider the ten numbers ( boldsymbol{a r}, boldsymbol{a r}^{2}, boldsymbol{a r} boldsymbol{3}, ldots ldots ldots boldsymbol{a} boldsymbol{r}^{10} ) If their sum is 18 and the sum of their reciprocal is 6 then the product of these ten numbers, is A ( cdot 3^{5} ) B . ( 3^{8} ) ( c cdot 3^{1} 0 ) D. ( 3^{1} 5 ) | 11 |

366 | The arithmetic mean of two numbers is 17 and their geometric mean is ( 15 . ) Find the numbers. | 11 |

367 | Is 51 a term of the ( A P 5,8,11,14, ldots ? ) | 11 |

368 | If the first term of a GP is 729 and ( 7^{t h} ) term is ( 64, ) then the sum of its first seven terms is A . 2187 в. 2059 ( c .1458 ) D. 2123 E . 1995 | 11 |

369 | The sum of the first 9 terms of an ( A . P ) is 81 and the sum of it’s first 20 terms is 400. Find the first term, the common difference and the sum upto 15 th term A ( cdot a=1, d=2, S_{15}=235 ) В . ( a=3, d=4, S_{15}=215 ) c. ( a=5, d=3, S_{15}=205 ) D. None of these | 11 |

370 | If ( 6^{t h} ) term of a G.P. is ( frac{1}{32} ) and ( 9^{t h} ) term is ( frac{1}{256}, ) then ( 11^{t h} ) term ( = ) A . 1024 в. ( frac{1}{1024} ) c. ( frac{1}{256} ) D. ( frac{1}{512} ) | 11 |

371 | If ( (boldsymbol{m}+mathbf{1})^{t h},(boldsymbol{n}+mathbf{1})^{t h} ) and ( (boldsymbol{r}+mathbf{1})^{t h} ) terms of an A.P. are in G.P and ( boldsymbol{m}, boldsymbol{n}, boldsymbol{r} ) are in H.P., then ratio of the first term of the A.P. to its common difference in terms of ( n ) is ( A cdot frac{n}{2} ) B. ( -frac{n}{2} ) ( c cdot frac{n}{3} ) D. ( -frac{n}{3} ) | 11 |

372 | Solve : ( frac{1}{1.2}+frac{1}{2.3}+ldots . .+frac{1}{n(n+1)}=? ) | 11 |

373 | Assertion If ( x_{1}>0, i=1,2,3, dots, 50 ) and ( sum_{i=1}^{50} x_{i}= ) ( mathbf{5 0} ) then minimum value of ( frac{mathbf{1}}{boldsymbol{x}_{mathbf{1}}}+frac{mathbf{1}}{boldsymbol{x}_{mathbf{2}}}+ ) ( +frac{1}{x_{50}} ) is 50 Reason ( boldsymbol{A} cdot boldsymbol{M} geq boldsymbol{G} boldsymbol{M} ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect | 11 |

374 | the sum of the first 25 terms of natural numbers A. 325 в. 675 ( c .625 ) D. 552 | 11 |

375 | Let ( A_{1}, G_{1} ) and ( H_{1} ) denote the arithmetic, geometric and harmonic means respectively of two distinct positive numbers. For ( n geq 2, ) let ( A_{n-1} ) and ( H_{n-1} ) have arithmetic, geometric and harmonic means as ( boldsymbol{A}_{n}, boldsymbol{G}_{n}, boldsymbol{H}_{n} ) respectively. Which of the following statements is correct? ( mathbf{A} cdot A_{1}>A_{2}>A_{3}>dots ) B . ( A_{1}<A_{2}<A_{3}A_{3}>A_{5} ldots ) and ( A_{2}<A_{4}<A_{6}<ldots ) D. ( A_{1}<A_{3}<A_{5}A_{4}>A_{6}>ldots . ) | 11 |

376 | The tenth term of the series 9,8,7,6 ( ldots . ) is | 11 |

377 | Find the first term and common difference of the A.P. (i) ( 5,2,-1,-4, dots ) (ii) ( frac{1}{2}, frac{5}{6}, frac{7}{6}, frac{3}{6}, dots, frac{17}{6} ) | 11 |

378 | How many terms are there in the sequence ( 3,6,9,12, ldots, 111 ? ) | 11 |

379 | Divide 96 into four parts which are in A.P. and the ratio between product of their means to product of their extremes is 15: 7 | 11 |

380 | The sum of the series ( frac{2}{3 !}+frac{4}{5 !}+frac{6}{7 !}+ldots ) ( infty=frac{a}{e} . ) Find ( (a+3)^{2} ) | 11 |

381 | 25. If (10)’ +2(11)’ (10%) +3(11)’ (10)’ +……… +10(11)’ = k (10), then k is equal to: (a) 100 (b) 110 @ 120 [JEE M 2014 441 (d) 100 10 | 11 |

382 | The sum of ( n ) terms of ( 1^{2}+left(1^{2}+right. ) ( left.2^{2}right)+left(1^{2}+2^{2}+3^{2}right)+dots . ) A ( cdot frac{n(n+a)(2 n+1)}{6} ) в. ( frac{n(n+1)(2 n-1)}{6} ) c. ( frac{1}{12} n(n+1)^{2}(n+2) ) D. ( frac{1}{12} n^{2}(n+1)^{2} ) | 11 |

383 | 9. Let T be the rth term of an A.P. whose first term is a and common difference is d. If for some positive integers men, m #n, Tm = – and Tn = —, then a-d equals 2004|| п m 1 1 (a) = + – (b) 10 (c) – (d) 0 m n mn | 11 |

384 | Sum the series to ( n ) terms ( frac{1}{2.4 .6}+frac{1}{4.6 .8}+frac{1}{6.8 .10}+dots ) | 11 |

385 | The sum of the first five terms and the sum of the first ten terms of an AP are same. Which one of the following is the correct statement? A. The first term must be negative B. The common difference must be negative c. Either the first term or the common difference is negative but not both D. Both the first term and the common difference are negative | 11 |

386 | In the given ( A P ), find 3 missing term : ( 2, dots, 26 ) | 11 |

387 | The first term of an arithmetic progression ( a_{1}, a_{2}, a_{3}_{-}-_{-}-_{-} ) is equal to unity then the value of the common difference of the progression if ( a_{1} a_{3}+ ) ( a_{2} a_{3} ) is minimum, is A. ( -5 / 4 ) B. – ( 5 / 2 ) ( c cdot 4 ) D. None of these | 11 |

388 | The sum of the first three terms of an A.P. is 9 and the sum of their square is 35. The sum to first ( n ) terms of the series can be This question has multiple correct options | 11 |

389 | An example of the odd number of terms in an arithmetic progression written in a symmetrical format is? A ( . a-d, a, a+d ) в. ( a, a+d, a+2 d ). c. ( a-2 d, a-d, a ) D. ( a-d, a+d, a+3 d ) | 11 |

390 | Which term of the sequence ( 114,109,104 dots dots ) is the first negative term | 11 |

391 | Sum ( 1, sqrt{3}, 3, dots ) to 12 terms. | 11 |

392 | The sum of the first ( n ) terms of the series ( 1^{2}+2 cdot 2^{2}+3^{3}+2 cdot 4^{2}+5^{2}+ ) ( 2 cdot 6^{2}+cdots ) is ( frac{n(n+1)^{2}}{2} ) when ( n ) is even when ( n ) is odd the sum is A ( cdot frac{3 n(n+1)}{2} ) B. ( frac{n^{2}(n+1)}{2} ) c. ( frac{n(n+1)^{2}}{4} ) D. ( left[frac{n(n+1)}{2}right]^{2} ) | 11 |

393 | Find the ( 9^{t h} ) term of a G.P.: ( 3,6,12,24, dots ) | 11 |

394 | If the number of shot in a triangular pile is to the number of shot in a square pile of double the number of courses as 13 to ( 175 ; ) find the number of shot in each pile. | 11 |

395 | Assume that ( a, b, c ) and ( d ) are positive integers such that ( a^{5}=b^{4}, c^{3}=d^{2} ) and ( c-a=19 . ) Determine ( d-b ) | 11 |

396 | If ( a, b, c, d ) are non-negative real numbers where ( a+b+c+d=1, ) then the maximum value of ( a b+b c+c d+ ) ( a d ) is A ( cdot frac{1}{4} ) B. 3 ( c cdot 4 ) D. none of these | 11 |

397 | If ( S=1^{2}-2^{2}+3^{2}-4^{2} dots ) upto ( n ) terms and ( n ) is even, then ( S ) equals A ( cdot frac{n(n+1)}{2} ) B. ( frac{n(n-1)}{2} ) c. ( frac{-n(n+1)}{2} ) D. ( frac{-n(n-1)}{2} ) | 11 |

398 | Write the value of ( x ) for which ( 2 x, x+10 ) and ( 3 x+2 ) are in ( A . P ) | 11 |

399 | The sum of three numbers which are consecutive terms of an A.P is ( 21 . ) If the second number is reduced by ( 1 & ) the third is increased by ( 1, ) we obtain three consecutive terms of a G.P., find the numbers. | 11 |

400 | Find the Odd one among : 23,13,34,25 56,51 A . 23 B. 13 ( c cdot 34 ) D. 51 | 11 |

401 | The ( 4 t h ) term from the end of the ( A P ) ( -11,-8,-5, dots dots dots dots ) A . 37 B . 40 ( c .43 ) D. 58 | 11 |

402 | ( ln ) G.P. ( (p+q) ) th term is ( m,(p-q) ) th term is n, then pth term is A . ( n m ) B. ( sqrt{n m} ) ( mathrm{c} cdot m / n ) D. ( sqrt{m / n} ) | 11 |

403 | The sum of an infinitely decreasing geometric progression is 1.5 and the sum of the squares of its terms is ( frac{1}{8} ) Find the progression. | 11 |

404 | Evaluate ( boldsymbol{S}=mathbf{1}+frac{mathbf{4}}{mathbf{5}}+frac{mathbf{7}}{mathbf{5}^{2}}+frac{mathbf{1 0}}{mathbf{5}^{3}}+dots dots ) to infinite terms. Find ( 16 S ) | 11 |

405 | Choose the correct answer from the alternatives given: The sum of ( frac{1}{sqrt{2}+1}+frac{1}{sqrt{3}+sqrt{2}}+ ) ( frac{1}{sqrt{4}+sqrt{3}}+ldots . .+frac{1}{sqrt{100}+sqrt{99}} ) is ( mathbf{A} cdot mathbf{9} ) B . 10 c. 11 D. None of these | 11 |

406 | If the sum to ( n ) terms of an A.P. is ( 3 n^{2}+ ) ( mathbf{5} n ) while ( T_{m}=164, ) then value of ( m ) is A . 25 B . 26 c. 27 D . 28 | 11 |

407 | If ( boldsymbol{A}_{boldsymbol{k}}=left[begin{array}{cc}boldsymbol{k} & boldsymbol{k}-mathbf{1} \ boldsymbol{k}-mathbf{1} & boldsymbol{k}end{array}right] ) then ( left|boldsymbol{A}_{mathbf{1}}right|+ ) ( left|boldsymbol{A}_{2}right|+ldots+left|boldsymbol{A}_{2015}right|=? ) ( A cdot 0 ) в. 2015 c. ( (2015)^{2} ) D. ( (2015)^{3} ) | 11 |

408 | If the ( p^{t h}, q^{t h} ) and ( r^{t h} ) term of a G.P. are a,b,c respectively, then ( a^{q-r} cdot b^{r-p} cdot c^{p-q} ) is equal to A. B. ( c cdot a b c ) D. par | 11 |

409 | Say true or false. A.M. of any ( n ) positive numbers ( a_{1}, a_{2}, a_{3}, dots, a_{n} ) is ( A ) then ( A= ) ( frac{boldsymbol{a}_{1}+boldsymbol{a}_{2}+ldots .+boldsymbol{a}_{boldsymbol{n}}}{boldsymbol{n}} ) A . True B. False | 11 |

410 | is a sum of numbers. A. sequence B. series ( c . ) term D. constant | 11 |

411 | If ( boldsymbol{alpha} inleft(mathbf{0}, frac{boldsymbol{pi}}{mathbf{2}}right) ) then ( sqrt{mathbf{x}^{2}+mathbf{x}}+frac{tan ^{2} boldsymbol{alpha}}{sqrt{mathbf{x}^{2}+mathbf{x}}} ) is always greater than or equal to ( A cdot 2 tan alpha ) B. ( c cdot 2 ) ( D cdot sec ^{2} alpha ) | 11 |

412 | The first and last term of an ( A . P . ) are ( a ) and ( l ) respectively. If ( S ) is the sum of all the terms of the ( A . P . ) and the common difference is ( frac{l^{2}-a^{2}}{k-(l+a)}, ) then ( k ) is equal to ( A ) B . ( 2 s ) ( c cdot 3 s ) D. None of these | 11 |

413 | The third term of an A.P. is ( 18, ) and the seventh term is ( 30 ; ) find the sum of 17 terms. | 11 |

414 | Second and fourth term of on A.P. is 12 and 20 respectively. Find the sum of first 25 terms of the A.P. | 11 |

415 | ( operatorname{Let} a_{1}, a_{2}, dots a_{3 n} ) be an arithmetic progression with ( a_{1}=3 ) and ( a_{2}=7 . ) If ( a_{1}+a_{2}+ldots+a_{3 n}=1830 ) then what is the smallest positive integer ( m ) such that ( boldsymbol{m}left(boldsymbol{a}_{1}+boldsymbol{a}_{2}+ldots+boldsymbol{a}_{n}right)>mathbf{1 8 3 0} ? ) | 11 |

416 | The number of the integers from 1 to 120 which are divisible by 3 or 5 A . 56 B. 40 ( c cdot 24 ) ( D cdot 8 ) | 11 |

417 | 1. The sum of integers from 1 to 100 that are divisible by 2 or 5 is ………. (1984 – 2 Marks) | 11 |

418 | Find the first term and the common difference of an ( A P, ) if the ( 3^{r d} ) term is 6 and the ( 1^{t h} ) term is 34 | 11 |

419 | Let ( A, G ) and ( H ) are the A.M., G.M. and H.M. respectively of two unequal positive integers. Then the equation ( A x^{2}- ) ( |G| x+frac{H}{4}=0 ) has This question has multiple correct options A. Both roots as fractions B. At least one root which is a negative fraction. c. Exactly one positive root D. At least one root which is an integer E. None of these. | 11 |

420 | Find the term ( t_{15} ) of an A.P.: ( 4,9,14, dots . ) | 11 |

421 | ( sum_{k=1}^{2 n+1}(-1)^{k-1} k^{2}= ) ( mathbf{A} cdot(n+1)(2 n+1) ) B. ( (n+1)(2 n-1) ) ( mathbf{c} cdot(n-1)(2 n-1) ) D. ( (n-1)(2 n+1) ) | 11 |

422 | If ( a_{r}>0 ; forall r, n in N ) and ( a_{1}, a_{2}, a_{3}, dots . a_{2 n} ) are in A.P, then ( frac{boldsymbol{a}_{mathbf{1}}+boldsymbol{a}_{mathbf{2} n}}{sqrt{boldsymbol{a}_{1}}+sqrt{boldsymbol{a}_{2}}}+frac{boldsymbol{a}_{mathbf{2}}+boldsymbol{a}_{mathbf{2} n-1}}{sqrt{boldsymbol{a}_{2}}+sqrt{boldsymbol{a}_{3}}}+ ) ( frac{boldsymbol{a}_{boldsymbol{3}}+boldsymbol{a}_{boldsymbol{2} boldsymbol{n}-boldsymbol{2}}}{sqrt{boldsymbol{a}_{boldsymbol{3}}}+sqrt{boldsymbol{a}_{boldsymbol{4}}}}+ldots+frac{boldsymbol{a}_{boldsymbol{n}}+boldsymbol{a}_{boldsymbol{n}+1}}{sqrt{boldsymbol{a}_{boldsymbol{n}}}+sqrt{boldsymbol{a}_{boldsymbol{n}+1}}}= ) ( mathbf{A} cdot n-1 ) в. ( frac{nleft(a_{1}+a_{2 n}right)}{sqrt{a_{1}}+sqrt{a_{n+1}}} ) c. ( frac{n-1}{sqrt{a_{1}}+sqrt{a_{n+1}}} ) D. ( n+1 ) | 11 |

423 | Solve : ( x+frac{1}{x} geq 2 ) в. ( R ) ( c cdot phi ) D. ( [0, infty) ) | 11 |

424 | The sum of first 45 natural numbers is: A. 1035 в. 1280 c. 2070 D. 2140 | 11 |

425 | In a geometric sequence, the is the ratio of a term to the previous term. A. constant B. common difference c. common ratio D. term | 11 |

426 | Given the sequence of numbers ( boldsymbol{x}_{1}, boldsymbol{x}_{2}, boldsymbol{x}_{3}, ldots ldots . . boldsymbol{x}_{2013} ) which satisfies ( frac{boldsymbol{x}_{mathbf{1}}}{boldsymbol{x}_{mathbf{1}}+mathbf{1}}=frac{boldsymbol{x}_{mathbf{2}}}{boldsymbol{x}_{mathbf{2}}+mathbf{3}}=frac{boldsymbol{x}_{mathbf{3}}}{boldsymbol{x}_{mathbf{3}}+mathbf{5}}=dots dots ) ( frac{x_{2013}}{x_{2013}+4025}, ) nature of the sequence is A . A.P в. ( G . P ) c. ( H . P ) D. A.G.P | 11 |

427 | Which term of an A.P : 21, 42, 63,…. is 210 ( ? ) A . gth B. 10th c. 12th D. 11th | 11 |

428 | The variance of the series ( boldsymbol{a}, boldsymbol{a}+boldsymbol{d}, boldsymbol{a}+boldsymbol{2} boldsymbol{d}, ldots ldots boldsymbol{a}+(boldsymbol{2} boldsymbol{n}-mathbf{1}) boldsymbol{d}, boldsymbol{a}+ ) 2nd is ( ^{text {A } cdot} frac{n(n+1)}{2} d^{2} ) ( ^{text {В }} cdot frac{n(n-1)}{6} d^{2} ) c. ( frac{n(n+1)}{6} d^{2} ) D. ( frac{n(n+1)}{3} d^{2} ) | 11 |

429 | Determine the value of ( k ) for which ( k^{2}+ ) ( 4 k+8,2 k^{2}+3 k+6 a n d 3 k^{2}+4 k+4 ) are in A.P. | 11 |

430 | The first three terms pf A.P are ( (3 y-1) cdot(3 y+5) ) and ( (5 y+1) . ) Then ( Y ) equals to | 11 |

431 | The series of natural numbers is divided into groups (1),(2,3,4),(3,4,5,6,7),(4,5,6,7,8,9 Let the sum of the numbers in ( n t h ) group be ( =[k n-m]^{2} ).FInd ( k+m ? ) | 11 |

432 | is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term. A . Geometric value B. Geometric series c. Arithmetic progression D. Arithmetic mean | 11 |

433 | If ( a, b, c, d ) are positive real numbers such that ( a+b+c+d=2 ) then ( M= ) ( (a+b)(c+d) ) satisfies the relation A. ( 0<M leq 1 ) в. ( 1 leq M leq 2 ) c. ( 2 leq M leq 3 ) D. ( 3 leq M leq 4 ) | 11 |

434 | Which of the following is not an example of a series? A ( .1,2,3,4,5,6, dots ) в. ( -2,0,2,4,6,8, ldots ) c. ( 1,1,2,3,5,8, . . ) D. None of the above | 11 |

435 | Consider the sequence ( 1,2,2,4,4,4,4,8,8,8,8,8,8,8,8, dots ) and so on. Then 1025 th terms will be A ( cdot 2^{9} ) B . ( 2^{11} ) ( c cdot 2^{10} ) D. ( 2^{12} ) | 11 |

436 | What is the common difference of an A.P. in which ( a_{24}-a_{17}=-28 ? ) | 11 |

437 | If ( sum_{n=1}^{5} frac{1}{n(n+1)(n+2)(n+3)}=frac{k}{3} ) then k is equal to? A ( frac{55}{336} ) в. ( frac{17}{105} ) c. ( frac{19}{112} ) D. | 11 |

438 | Find ( 10 t h ) and 16 th terms of the G.P. ( 256,128,64, dots dots ) | 11 |

439 | The first term of an AP is 148 and the common difference is ( -2 . ) If the AM of first ( n ) terms of the ( A P ) is ( 125, ) then the value of ( n ) is A . 18 B . 24 c. 30 D. 36 E . 48 | 11 |

440 | Example 3. in the following ( A P_{S} ) find teh ,missing terms in the boxes: A. ( 2,, 26 ) B . ( 5, quad,, 9 frac{1}{2} ) ( mathrm{c} ldots, 13,, 3 ) D. Do all the above options | 11 |

441 | Say true or false. Zero can be the common ratio of a G.P. A . True B. False | 11 |

442 | Find the sum of the following AP ( 1,3,5,7 dots dots dots dots .199 ) | 11 |

443 | The first and last terms of an A.P of ( n ) terms is 1,31 respectively. The ratio of ( 8^{t h} ) term and ( (n-2)^{t h} ) term is ( 5: 9, ) the value of ( n ) is: ( A cdot 14 ) B. 15 ( c cdot 16 ) D. 13 | 11 |

444 | Find the common difference and write the next four terms of the ( A P: ) ( -1, frac{1}{4}, frac{3}{2}, ) A ( cdot d=frac{5}{2} ; a_{4}=frac{11}{4}, a_{5}=frac{16}{4}, a_{6}=frac{21}{4}, a_{7}=frac{26}{4} ) B. ( d=frac{5}{4} ; a_{4}=frac{11}{4}, a_{5}=frac{16}{4}, a_{6}=frac{21}{4}, a_{7}=frac{26}{4} ) C. ( d=frac{5}{4} ; a_{4}=frac{11}{4}, a_{5}=frac{16}{4}, a_{6}=frac{21}{4}, a_{7}=frac{25}{4} ) D. ( d=frac{5}{2} ; a_{4}=frac{11}{4}, a_{5}=frac{16}{4}, a_{6}=frac{21}{4}, a_{7}=frac{25}{4} ) | 11 |

445 | ( S_{r} ) denotes the sum of the first ( r ) terms of an AP. Then ( S_{3 n}:left(S_{2 n}-S_{n}right) ) is This question has multiple correct options A . ( n ) B. ( 3 n ) ( c .3 ) D. independent of ( n ) | 11 |

446 | If the sum of first 8 and 19 terms of an A.P. are 64 and 361 respectively, find the common difference. | 11 |

447 | Sum the following series to n terms and to infinity ( frac{1}{1.2}+frac{1}{2.3}+frac{1}{3.4}+dots . ) | 11 |

448 | f ( x, y, z ) are three real numbers of the same sign, then the value of ( left(frac{x}{y}+frac{y}{z}+frac{z}{x}right) ) lies in the interval A. ( [2, infty) ) (n) B. ( [3, infty) ) ( c cdot(3, infty) ) D. ( (-infty, 3) ) | 11 |

449 | The sum of three consecutive terms of an ( A P ) is 21 and the sum of the squares of these terms is 165: Find these terms | 11 |

450 | The sum of ( n ) terms of an ( A P ) is ( 3 n^{2}+5 n ) find the AP. Hence find its ( 16^{t h} ) term. | 11 |

451 | ( ln mathbf{A P} a_{3}=mathbf{2 1}, boldsymbol{a}, mathbf{3}=mathbf{2 1}, boldsymbol{a}_{mathbf{1}} mathbf{0}=mathbf{6 3} ) find ( boldsymbol{a}_{mathbf{5 0}} ) | 11 |

452 | If ( x^{2}+frac{1}{x^{2}}=A ) and ( x-frac{1}{x}=B ) then least value of ( frac{boldsymbol{A}}{boldsymbol{B}} ) is A .2 B. ( sqrt{2} ) ( c cdot-sqrt{2} ) D. ( 2 sqrt{2} ) | 11 |

453 | If ( beta neq 1 ) be any nth root of unity then prove that ( 1+3 beta+5 beta^{2}+ldots .+ ) ( n ) terms ( =-frac{2 n}{1-beta} ) | 11 |

454 | Find four numbers in G.P in which the sum of the extreme terms is 112 and sum of middle terms is 48 . | 11 |

455 | The sum of the first ( n ) terms of an ( A P ) is ( 3 n^{2}+6 n . ) Find the ( n^{t h} ) term of this ( A . P ) | 11 |

456 | Find two numbers whose arithmetic mean is 34 and the geometric mean is 16 | 11 |

457 | Between 1 and ( 31, m ) arithmetic means are inserted, so that the ratio of the ( 7^{t h} ) and ( (m-1)^{t h} ) mean is ( 5: 9 . ) Then the value of ( m ) is A . 12 B. 13 c. 14 D. 15 | 11 |

458 | What is the missing number in the sequence ( 1,5,10,16,23,31,_{—} ) A . 37 B. 38 c. 39 D. 40 E. 41 | 11 |

459 | ( boldsymbol{a}_{boldsymbol{n}}=frac{boldsymbol{n}(boldsymbol{n}-boldsymbol{2})}{boldsymbol{n}+boldsymbol{3}} ; boldsymbol{a}_{20} ) | 11 |

460 | If you have a finite arithmetic sequence, the first number is 2 and the common difference is ( 4, ) what is the ( 5^{t h} ) number in the sequence? ( mathbf{A} cdot mathbf{6} ) B. 10 c. 14 D. 18 | 11 |

461 | ( boldsymbol{S}=boldsymbol{3}^{10}+boldsymbol{3}^{9}+frac{3^{9}}{4}+frac{3^{7}}{2}+frac{5.3^{6}}{16}+frac{3^{2}}{16}+ ) ( frac{7.3^{4}}{64}+ldots ldots ldots ) upto infinite terms, then ( left(frac{25}{36}right) S ) equal to A ( cdot 6^{text {? }} ) B. ( 3^{10} ) ( c cdot 3^{1} ) D. ( 2.3^{10} ) | 11 |

462 | f ( a, b, c ) are in ( G . P, ) then ( aleft(b^{2}+c^{2}right)=cleft(a^{2}+b^{2}right) ) A. True B. False | 11 |

463 | The G.M. of n positive terms ( boldsymbol{x}_{1}, boldsymbol{x}_{2}, ldots . . boldsymbol{x}_{n} ) is A ( cdotleft(x_{1} times x_{2} times ldots ldots times x_{n}right)^{n} ) B ( cdot frac{1}{n}left(x_{1} times x_{2} times ldots times x_{n}right) ) c. ( left(x_{1} times x_{2} times ldots times x_{n}right)^{1 / n} ) D. None of these | 11 |

464 | Three numbers are in arithmetic progression. Their sum is 21 and the product of the first number and the third number is ( 45 . ) Then the product of these three number is A . 315 B. 90 ( c .180 ) D. 270 | 11 |

465 | If the first term of G.P. is ( 7, ) its ( n^{t h} ) term is 448 and sum of first ( n ) terms is 889 then find the fifth term of G.P. | 11 |

466 | For an A.P., find ( S_{7} ) if ( a=5 ) and ( d=4 ) | 11 |

467 | Find the value of ( k ) for which ( k, 2 k-1 ) and ( 2 k+1 ) are in ( A . P ) | 11 |

468 | If ( n^{t h} ) term of AP is ( t_{n}=4 n+1 . ) Find mean of first 10 terms. A . 85 B. 95 ( c cdot 23 ) D. 7.5 | 11 |

469 | ( fleft(a_{1}, a_{2}, a_{3}, dots, a_{n-1}, a_{n} ) are in A.P., then right. show that ( frac{1}{a_{1} a_{n}}+frac{1}{a_{2} a_{n-1}}+frac{1}{a_{3} a_{n-2}}+ ) ( ldots cdot frac{1}{a_{n} a_{1}}= ) ( frac{2}{left(a_{1}+a_{n}right)}left[frac{1}{a_{1}}+frac{1}{a_{2}}+dots frac{1}{a_{n}}right] ) | 11 |

470 | The arithmetic mean of first ten natural numbers is A . 5.5 B. 6 ( c .7 .5 ) D. 10 | 11 |

471 | Write the next four terms of the following A.P. ( frac{1}{6}, frac{1}{3}, frac{1}{2} ) | 11 |

472 | Find ( a_{30} ) given that the first few terms of a geometric sequence are given by ( -2,1,-frac{1}{2}, frac{1}{4} dots ) A ( cdot frac{1}{2^{27}} ) в. ( frac{1}{2^{2}} ) c. ( frac{1}{2^{28}} ) D. ( frac{1}{2^{29}} ) | 11 |

473 | Write the expression for the common difference of an ( A . P ) whose first term is ( a ) and ( n t h ) term is ( b ) | 11 |

474 | If the roots of the equation ( 4 x^{3}-12 x^{2}+ ) ( 11 x+k=0 ) are in A.P. Then ( k= ) A . -3 B. ( c cdot 2 ) D. 3 | 11 |

475 | If ( x_{1}, x_{2}, x_{3}, x_{4} ) are in G.P then its common ratio is, ( ^{mathbf{A}} cdotleft(frac{a r}{c p}right)^{frac{1}{4}} ) ( ^{mathrm{B}}left(frac{c r}{a p}right)^{frac{1}{3}} ) ( ^{c} cdotleft(frac{c r}{a p}right)^{frac{1}{2}} ) ( ^{mathrm{D}}left(frac{a p}{b q}right)^{frac{1}{2}} ) | 11 |

476 | Let ( S_{n} ) denote the sum of first ( n ) terms of an ( A P ) and ( 3 S_{n}=S_{2 n} . ) What is ( S_{3 n}: S_{n} ) equal to? A . 4: 1 B. 6: 1 c. 8: 1 D. 10: 1 | 11 |

477 | If ( H ) is harmonic mean between ( P ) and ( Q . ) Then the value of ( frac{boldsymbol{H}}{boldsymbol{P}}+frac{boldsymbol{H}}{boldsymbol{Q}} ) is ( A cdot 2 ) в. ( frac{P Q}{P+Q} ) c. ( frac{P+Q}{P Q} ) D. None of these | 11 |

478 | If the sum of three numbers in A.P. is 24 and their product is ( 440, ) find the numbers. | 11 |

479 | If the Arithmetic Mean of two numbers is twice their Geometric Mean, then the ratio of larger number to the smaller number is This question has multiple correct options A. ( 7-4 sqrt{3}: 1 ) B. ( 7+4 sqrt{3}: 1 ) c. 21: 1 D. 5: 1 | 11 |

480 | Which term of ( G . P .: 3,9,27, ldots . . ) is ( mathbf{2 1 8 7} ? ) | 11 |

481 | The product of three consecutive even numbers when divided by 8 is ( 720 . ) The product of their square root is : A. ( 12 sqrt{10} ) (1) 10111 B. 24sqrt10 ( c cdot 120 ) D. None of these | 11 |

482 | The sum of two numbers is ( 2 frac{1}{6} . ) An even number of arithmetic means are being inserted between them and their sum exceeds their number by 1. Find the number of means inserted. | 11 |

483 | Identify the geometric series. A. ( 1+3+5+7+dots ) B. ( 2+12+72+432 ) c. ( 2+3+4+5+dots ) D. ( 11+22+33+44+ldots ) | 11 |

484 | Find three consecutive even number whose sum is 234 | 11 |

485 | Let ( left(1+x^{2}right)^{2} cdot(1+x)^{n}=sum_{K=0}^{n+4} a_{K} cdot x^{K} ) ( fleft(a_{1}, a_{2} text { and } a_{3} text { are in A.P, find } nright. ) | 11 |

486 | If the sum of the infinity of the series ( 3+5 r+7 r^{2}+dots ) is ( frac{44}{9}, ) then find the value of ( r ) | 11 |

487 | How many terms of the series ( 12,16,20, dots . . ) must be taken to make the sum equal to ( 208 ? ) | 11 |

488 | The 3 rd and 6 th term of an arithmetic progression are 13 and -5 respectively. What is the 1 1th term? A . -29 в. -41 c. -47 D. -35 | 11 |

489 | O 3. Ifin a triangle PQR, sin P, sin ,sin Rare in A.P., then (1998 – 2 Marks) (a) the altitudes are in A.P. (b) the altitudes are in H.P. © the medians are in G.P. (d) the medians are in A.P. | 11 |

490 | The sum of first ( n ) terms of ( A . P . ) is ( 3 n+ ) ( n^{2} ) then Find second, third and ( 15^{t h} ) term. This question has multiple correct options A. 6 B. 8 c. 32 D. 45 | 11 |

491 | f ( t_{n}=4 n-3, ) then find the first two terms of an A.P. | 11 |

492 | The first term of the G.P. is 25 and 6 th term is ( 800 . ) Find the seventh term. | 11 |

493 | Write the common difference of an A.P. whose ( n^{t h} ) terms is ( 3 n+5 ) | 11 |

494 | The nth triangular number is defined to be the sum of the first ( n ) positive integers. For example, the ( 4^{t h} ) triangular number is ( 1+2+3+4=10 . ) In the first 100 terms of the sequence 1,3,6,10,15,21,28 of triangular numbers, how many are divisible by ( 7 ? ) | 11 |

495 | How many terms of the A.P. ( 1,4,7, ldots ) are needed to make the sum ( 51 ? ) | 11 |

496 | f ( x ) is the ( n^{t h} ) term of the ( A P ) ( 5 frac{1}{2}, 11,16 frac{1}{2}, 22 ldots ldots ) with ( x=550 ) find ( ^{prime} boldsymbol{n}^{prime} ) | 11 |

497 | Sum the following series to n terms and o infinity ( frac{1}{1.3 .5}+frac{1}{3.5 .7}+frac{1}{5.7 .9}+ldots . ) | 11 |

498 | A strain of bacteria doubles in numbers each day, if today there are 16 million bacteria, how many days ago were there 500,000 bacteria? | 11 |

499 | The ratio between the sum of ( n ) terms of two A.P.’s is ( 3 n+8: 7 n+15 . ) Find the ratio between their 12 th terms. A ( cdot frac{7}{10} ) B. ( frac{7}{18} ) c. ( frac{7}{12} ) D. ( frac{7}{16} ) | 11 |

500 | if ( a, b, c ) are distinct and the roots of ( (b-c) x^{2}+(c-a) x+(a+b)=0 ) are equal, then ( a, b, c ) are in A. Arithmetic progression B. Geometric progression c. Harmonic progression D. Arithmetico-Geometric progression | 11 |

501 | Find ( a_{1}+a_{6}+a_{11}+a_{16} ) if it is known that ( a_{1}, a_{2} dots ) is an A.P. and ( a_{1}+a_{4}+ ) ( boldsymbol{a}_{boldsymbol{7}}+ldots+boldsymbol{a}_{mathbf{1 6}}=mathbf{1 4 7} ) | 11 |

502 | If the roots of the equation ( x^{3}-12 x^{2}+ ) ( 39 x-28=0 ) are in A.P., then their common difference will be, | 11 |

503 | Find the nth term of the sequence ( 0.7,0.77,0.777 ldots ) | 11 |

504 | 10. Let p be the first ot be the first of the n arithmetic means between two umbers and q the first of n harmonic means between the me numbers. Show that q does not lie between p and Sam (n+12 (1991 – 4 Marks) (n-1 | 11 |

505 | If the ( p^{t h} ) term is ( q ) and ( q^{t h} ) term is ( p ) of an A.P., then find the first term and common difference. | 11 |

506 | The ( n^{t h} ) term of a Geometric Progression is ( a_{n}=a r^{n-1}, ) where ( r ) represents A. Common difference B. Common ratio c. First term D. Radius | 11 |

507 | Three vertical poles of heights ( h_{1}, h_{2} ) and ( h_{3} ) at the vertices ( A, B ) and ( C ) of ( a ) ( triangle A B C ) subtend angles ( alpha, beta, gamma ) respectively at the circumcentre of the triangle. If ( cot alpha, cot beta, cot gamma ) are in ( A P ) then ( h_{1}, h_{2}, h_{3} ) are in ( A cdot A P ) в. GР c. нр D. None of these | 11 |

508 | s is the sum to infinite terms of a G.P. whose first term is 1. Then the sum of ( n ) term is A ( cdot sleft(1-left(1-frac{1}{s}right)^{n}right) ) B ( cdot frac{1}{s}left(1-left(1-frac{1}{s}right)^{n}right) ) c. ( _{1-}left(1-frac{1}{s}right)^{n} ) D. ( 1+left(1-frac{1}{s}right)^{n} ) | 11 |

509 | ( boldsymbol{a}^{boldsymbol{x}}=boldsymbol{b}, boldsymbol{b}^{boldsymbol{y}}=boldsymbol{c}, boldsymbol{c}^{boldsymbol{z}}=boldsymbol{a} ) Find the value of ( x, y, z ) ( mathbf{A} cdot mathbf{1} ) B. Not valid c. -1 D. | 11 |

510 | Find five numbers in Arithmetic progression whose sum is 25 and the sum of whose squares as 135 | 11 |

511 | If ( x neq y ) and the sequences ( x, a_{1}, a_{2}, y ) and ( x, b_{1}, b_{2}, y ) each are in ( A . P ., ) then ( left(frac{a_{2}-a_{1}}{b_{2}-b_{1}}right) ) A ( cdot frac{2}{3} ) B. ( frac{3}{2} ) c. 1 D. | 11 |

512 | If ( a^{2}(b+c), b^{2}(c+a), c^{2}(a+b) ) are in AP, then ( a, b, c ) are in ( A cdot A P ) в. GР ( c . ) не D. None of these | 11 |

513 | If the first term in a geometric sequence is ( 3, ) and if the third term is ( 48, ) find the ( 11^{t h} ) term. A . 228 в. 528 c. 110592 D. 3145728 E . 12582912 | 11 |

514 | Find the sum to n terms of the sequence, ( 8,88,888,8888, dots ) | 11 |

515 | If ( boldsymbol{T}_{boldsymbol{n}}=boldsymbol{6} boldsymbol{n}+mathbf{5}, ) find ( boldsymbol{S}_{boldsymbol{n}} ) | 11 |

516 | If ( S_{r} ) denotes the sum of the first ( r ) terms of an ( A P ) then ( frac{S_{3 r}-S_{r-1}}{S_{2 r}-S_{2 r-1}} ) is equal to A ( .2 r-1 ) в. ( 2 r+1 ) c. ( 4 r+1 ) D. ( 2 r+3 ) | 11 |

517 | Coefficient of ( boldsymbol{x}^{r} ) in ( mathbf{1}+(mathbf{1}+boldsymbol{x})+(mathbf{1}+ ) ( x)^{2}+ldots ldots+(1+x)^{n} ) is A. ( ^{n+3} C_{r} ) B. ( ^{n+1} C_{r+1} ) ( mathbf{c} cdot^{n} C_{r} ) D. ( (n+2) C_{r} ) | 11 |

518 | find the sum of the first 20 even natural numbers. A . 400 в. 410 c. 420 D. 430 | 11 |

519 | The sum of first 20 terms of ( boldsymbol{A P} ) : ( mathbf{8}, mathbf{3},-mathbf{2}, dots . . ) is: ( mathbf{A} cdot-790 ) B. -970 ( mathbf{c} .-979 ) D. -779 | 11 |

520 | Three numbers whose sum is 15 are in A.P. If 1,4,19 be added to them respectively, then they are, in ( G . P . ) Find the numbers. | 11 |

521 | Let ( S_{n} ) denote the sum of first ( n ) terms of an ( A P, ) If ( S_{2 n}=3 S_{n} ) then find the ratio ( boldsymbol{S}_{boldsymbol{3} boldsymbol{n}} / boldsymbol{S}_{boldsymbol{n}} ) | 11 |

522 | The mean of ( x, y, z ) is ( y, ) then ( x+z= ) A ( . y ) в. ( 3 y ) c. ( 2 y ) D. ( 4 y ) | 11 |

523 | Evaluate ( sum_{k=1}^{11}left(2+3^{k}right) ) | 11 |

524 | ( ln ) an A.P., if ( a=3.5, d=0, n=101 ) then ( a_{n} ) will be ( mathbf{A} cdot mathbf{0} ) в. 3. c. 103.5 D. 104.5 | 11 |

525 | ( a_{n}=(-1)^{n-1} n^{3} ; a_{9} ) | 11 |

526 | If ( a^{2}, b^{2}, c^{2} ) are in A.P. then the following are also in A.P. True or False? If true write 1 otherwise write 0 ( frac{1}{b+c}, frac{1}{c+a}, frac{1}{a+b} ) | 11 |

527 | If ( lim _{x rightarrow 0^{+}} xleft(left[frac{1}{x}right]+left[frac{5}{x}right]+left[frac{11}{x}right]+left[frac{19}{x}right]+right. ) 430 (where [.] denotes the greatest integer function), then ( n= ) A . 8 B. 9 ( c .10 ) D. 11 | 11 |

528 | Which of the following are ( A p s ? ) If they form an ( A P, ) find the common difference ( d ) and write three more terms. ( 2, frac{5}{2}, 3 frac{7}{2}, dots ) | 11 |

529 | Show for positive number ( a, b, c frac{b c}{a}+ ) ( frac{a c}{b}+frac{a b}{c} leq a+b+c ) | 11 |

530 | Which term of the G.P. ( 2,2 sqrt{2}, 4, dots ) is 32? | 11 |

531 | The product of the third by the sixth term of an arithmetic progression is 406. The division of the ninth term of the progression by the fourth term gives a quotient 2 and a remainder ( 6 . ) Find the first term and the difference of the progression. | 11 |

532 | If ( S_{n}=sum_{r=1}^{n} frac{2 r+1}{r^{4}+2 r^{3}+r^{2}} ) then ( S_{20}= ) A ( cdot frac{220}{2210} ) в. ( frac{420}{44} ) c. ( frac{439}{221} ) D. ( frac{440}{441} ) | 11 |

533 | If ( A M ) and ( G M ) of two positive numbers a and b are 10 and 8 respectively,find the numbers | 11 |

534 | If ( G ) is the geometric mean of ( x ) and ( y ) then ( t frac{1}{G^{2}-x^{2}}+frac{1}{G^{2}-y^{2}}=frac{1}{G^{2}} ) | 11 |

535 | A.P. ( mathbf{A} cdot 2 a_{2}^{3}-3 a_{0} a_{1} a_{2}+a_{0}^{2} a_{3}=0 ldots(1) ) B ( cdot 2 a_{2}^{3}-3 a_{0} a_{1} a_{2}+a_{0}^{2} a_{1}=0 ldots(1) ) C ( cdot 2 a_{1}^{3}-3 a_{0} a_{1} a_{2}+a_{0}^{2} a_{3}=0 ldots(1) ) D. ( 2 a_{1}^{3}-3 a_{0} a_{1} a_{2}+a_{0}^{2} a_{1}=0 ldots(1) ) | 11 |

536 | Let ( m ) and ( n(m<n) ) be the roots of the equation ( boldsymbol{x}^{2}-mathbf{1 6 x}+mathbf{3 9}=mathbf{0 .} ) If four terms ( p, q, r ) and ( s ) are inserted between ( m ) and ( n ) form an ( A P, ) then what is the value of ( boldsymbol{p}+boldsymbol{q}+boldsymbol{r}+boldsymbol{s} ? ) A . 29 B. 30 ( c cdot 32 ) D. 35 | 11 |

537 | The arithmetic mean of the nine numbers in the given set {9,99,999,999999999} is a 9 digit numbers ( N, ) all whose digits are distinct. The number ( N ) does not contain the digit | 11 |

538 | The common difference of the AP ( frac{1}{P}, frac{1-P}{P}, frac{1-2 P}{P}, ldots ) ( A cdot P ) ( B .-1 ) ( c .- ) ( D ) | 11 |

539 | The first term of an A.P. is 2 and ninth term is ( 58 . ) What is the common difference? A . 5 B. 6 ( c cdot 7 ) D. | 11 |

540 | What does the series ( 1+3^{frac{-1}{2}}+3+ ) ( frac{1}{3 sqrt{3}}+ldots . ) represent? ( A cdot A P ) в. GP c. нр D. None of the above series | 11 |

541 | Find the first term of the sequence whose ( n^{t h} ) is given as: ( boldsymbol{t}_{boldsymbol{n}}=mathbf{4} boldsymbol{n}-boldsymbol{3} ) A .2 B. 1 ( c .3 ) D. 4 | 11 |

542 | The ratio of 7 th to the ( 3 r d ) term of an A.P is ( 12: 5 . ) Find the ratio of 13 th to the ( 4 t h ) term. | 11 |

543 | If ( alpha, beta ) are the roots of the equation ( x^{2}- ) ( 4 x+lambda=0 ) and ( gamma, delta ) are the roots of the equation ( boldsymbol{x}^{2}-mathbf{6 4} boldsymbol{x}+boldsymbol{mu}=mathbf{0} ) and ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma}, boldsymbol{delta} ) are given to be in increasing G.P. The value of ( mu / lambda ) be ( 2^{k} . ) Find ( k ? ) | 11 |

544 | In a row of girls, Mridula is ( 18^{t h} ) from the right and Sanjana is ( 18^{t h} ) from the left. If both of them interchange their position, Sanjana becomes ( 25^{t h} ) from the left, how many girls are there in the row? A . 40 B. 41 c. 42 D. 35 5 | 11 |

545 | Find the greatest value of ( x y z ) for positive value of ( x, y, z ) subject to the condition ( boldsymbol{x} boldsymbol{y}+boldsymbol{y} boldsymbol{z}+boldsymbol{z} boldsymbol{x}=mathbf{1 2} ) | 11 |

546 | The ( 9^{t h} ) term of an A.P.is equal to 6 times its second term. If its ( 5^{t h} ) term is ( 22, ) then find the A.P. | 11 |

547 | The common ratio is calculated in A . A.P. в. G.P. c. н.P. D. I.P. | 11 |

548 | The geomatric mean of 4,6,8 A ( cdot 2 sqrt[3]{6} ) B. ( sqrt{24} ) ( c cdot 6 sqrt{2} ) D. ( 4 sqrt[3]{3} ) | 11 |

549 | If the sum of ( m ) terms of an arithmetical progression is equal to the sum of either the next ( n ) terms or the next ( p ) terms, then ( left(frac{n+m}{n-m}right)left(frac{p-m}{p+m}right) ) is A ( cdot frac{n}{p} ) в. ( frac{p}{n} ) c. ( n p ) D. ( frac{p}{m} ) | 11 |

550 | Find an A.P. in which sum of any number of terms is always three times the squared number of these terms. | 11 |

551 | Find the sum to n terms of the sequence ( 8,88,888,8888 dots ) | 11 |

552 | If ( 2 p+3 q+4 r=15 ) then the maximum value of ( p^{3} q^{5} r^{7} ) is? A . 2180 в. ( frac{5^{4} cdot 3^{5}}{2^{15}} ) c. ( frac{5^{5} cdot 7^{7}}{2^{17} cdot 9} ) D. 2285 | 11 |

553 | If ( a, b, c ) and ( d ) are in harmonic progression, then ( frac{1}{a}, frac{1}{b}, frac{1}{c} ) and ( frac{1}{d}, ) are in progression. ( A cdot A P ) в. GP c. нр D. AGP | 11 |

554 | If ( frac{1}{1 ! 10 !}+frac{1}{2 ! 9 !}+frac{1}{3 ! 10 !}+ldots+ ) ( frac{1}{1 ! 10 !}=frac{2}{k !}left(2^{k-1}-1right) ) then find the value of k. | 11 |

555 | Find the last term of the following sequence ( 2,3,4 frac{1}{2} ldots ldots ) to 6 terms | 11 |

556 | Find the Odd one among : 35,63,105,121 133,210 A . 35 B. 63 ( c cdot 105 ) D. 121 | 11 |

557 | f ( log _{3} 2, log _{3}left(2^{x}-5right) ) and ( log _{3}left(2^{x}-right. ) ( 7 / 2) ) are in A.P, then ( x ) is equal to ( A cdot 2 ) B. 3 ( c cdot 4 ) D. 2,3 | 11 |

558 | If ( sqrt{2}+sqrt{6}+sqrt{18}+ldots . .10 ) terms is equal to ( k(sqrt{2}+sqrt{6}), ) then find the value of k. | 11 |

559 | ( mathbf{3}+mathbf{3 3}+mathbf{3 3 3}+ldots . .+mathbf{n} ) terms. | 11 |

560 | If AM of two numbers is twice of their GM, then the ratio of greatest number to smallest number is A. ( 7-4 sqrt{3} ) B. ( 7+4 sqrt{3} ) ( c cdot 2 ) D. 5 | 11 |

561 | For the following arithmetic progressions write the first term ( a ) and the common difference ( boldsymbol{d} ) ( -5,-1,3,7, dots ) | 11 |

562 | If the Arithmetic mean of ( 8,6,4, x, 3,6,0 ) is ( 4 ; ) then the value of ( boldsymbol{x}= ) ( A cdot 7 ) B. 6 ( c .1 ) D. 4 | 11 |

563 | Show that the products of the corresponding terms of the sequences ,ar ( , a r^{2}, a r^{n-1}, ) and ( A ) from a G.P. and find the common ratio. | 11 |

564 | ff ( p ) th ( , q ) th ( , r ) th and ( s ) th terms of a G.p,prove that; ( (p-q),(q-r) ) and ( (r-s) ) are also in G.P. | 11 |

565 | Write the first five fterms of the following square or find obtain the corresponding series: ( boldsymbol{C} boldsymbol{I}_{1}=-mathbf{1}, boldsymbol{a} boldsymbol{n}= ) ( frac{a n-1}{n}, n geq 2 ) | 11 |

566 | If three terms are in A.P. and there sum is ( 18 . ) Find the second term. | 11 |

567 | Find whether 0 is a term of the A.P. ( mathbf{4 0}, mathbf{3 7}, mathbf{3 4}, mathbf{3 1} dots ) | 11 |

568 | In a geometrical progression first term and common ratio are both ( frac{1}{2}(sqrt{3}+i) ) Then the absolute value of the nth term of the progression is ( mathbf{A} cdot 2^{n} ) B . ( 4^{n} ) ( c ) D. none of these | 11 |

569 | In an arithmetic series, ( a_{1}=7 ) and ( a_{12}=29 . ) Find the sum of the first 12 terms. A . 116 в. 216 ( c cdot 316 ) D. 416 | 11 |

570 | The common ratio of ( mathrm{GP} mathbf{4}, mathbf{8}, mathbf{1 6}, mathbf{3 2}, dots . ) is ( A cdot 2 ) B. 3 ( c cdot 4 ) ( D ) | 11 |

571 | ( mathbf{3 . 6}+mathbf{6 . 9}+mathbf{9 . 1 2}+ldots+mathbf{3 n}(mathbf{3 n}+mathbf{3})= ) A ( frac{n(n+1)(n+2)}{3} ) B. ( 3 n(n+1)(n+2) ) c. ( frac{(n+1)(n+2)(n+3)}{3} ) ( frac{(n+1)(n+2)(n+4)}{4} ) | 11 |

572 | Which term of an AP 3,15,27,39 will be 120 more than its ( 21^{s t} ) term | 11 |

573 | ( mathbf{1 5}, mathbf{1 0}, mathbf{5}, dots dots dots ) In this A.P., sum of first 10 terms is | 11 |

574 | For the following geometric progression find the ( n^{t h} ) term ( 2,6,18,54, dots dots ) | 11 |

575 | If In(a + c), In (a-c), In (a-2b + c) are in A.P., then (1994) (a) a, b, care in A.P. (b) a, b2, c2 are in A.P. (c) a, b, c are in G.P. (d) a, b, c are in H.P. | 11 |

576 | If ( 8^{t h} ) term of an A.P is 15 , then the sum of 15 terms is A . 15 B. 0 c. 225 D. ( frac{225}{2} ) | 11 |

577 | If each observation is multiplied by ( frac{1}{3} ) then the mean of the new data will de A ( cdot frac{1}{3} ) times B. 3 times c. ( frac{1}{sqrt{3}} ) times D. ( frac{2}{3} ) times | 11 |

578 | The difference between the first and the fifth term of a geometric progression whose all terms are positive numbers is 15 and the sum of the first and the third term of the progression is ( 20 . ) Calculate the sum of the first five terms of the progression. | 11 |

579 | Find the nth term of the A.P.’s: 2, 7, 12, 17, | 11 |

580 | The nth term of the geometric progression ( -3,6,-12,24 dots ) is A. ( -3(-2)^{n+1} ) B. ( 3(-2)^{n-1} ) c. ( -3(2)^{n-1} ) D. ( -3(-2)^{n-1} ) | 11 |

581 | The sum ( mathbf{V}_{mathbf{1}}+mathbf{V}_{mathbf{2}}+ldots+mathbf{V}_{mathbf{n}} ) is A ( cdot frac{1}{12} mathrm{n}(mathrm{n}+1)left(3 mathrm{n}^{2}-mathrm{n}+1right) ) B. ( frac{1}{12} mathrm{n}(mathrm{n}+1)left(3 mathrm{n}^{2}+mathrm{n}+2right) ) C ( cdot frac{1}{2} mathrm{n}left(2 mathrm{n}^{2}-mathrm{n}+1right) ) D ( cdot frac{1}{3}left(2 mathrm{n}^{3}-2 mathrm{n}+3right) ) | 11 |

582 | For given A.P. ( -frac{1}{2},-frac{3}{2}, frac{1}{2},-frac{3}{2}, . . ) find the common difference. A . -1 B. ( -frac{1}{2} ) ( c cdot frac{3}{2} ) ( D ) | 11 |

583 | Find the sum to ( n ) terms of the series ( mathbf{1} cdot mathbf{2}^{2}+mathbf{2} cdot mathbf{3}^{2}+mathbf{3} cdot mathbf{4}^{2}+dots ) | 11 |

584 | If ( x^{2}+frac{1}{x^{2}}=A ) and ( x-frac{1}{x}=B, ) then least value of ( frac{mathbf{A}}{B} ) is A .2 B. ( sqrt{2} ) ( c cdot-sqrt{2} ) D. ( 2 sqrt{2} ) | 11 |

585 | Find ( S_{n} ) for each of the geometirc series described below. (i) ( a=3, t_{8}=384, n=8 ) (ii) ( a=5, r=3, n=12 ) | 11 |

586 | f ( a, b, c ) are in A.P., prove that the following are also in A.P. ( frac{1}{sqrt{(b)}+sqrt{(c)}}, frac{1}{sqrt{(c)}+sqrt{(a)}}, frac{1}{sqrt{(a)}+v} ) | 11 |

587 | 11. Let a, ß be the roots of x2 – x+p=0 and y, 8 be the roots of x2 – 4x +q=0. Ifa, ß, y, d are in GP., then the integral values of p and q respectively, are (20015) (a) -2, -32 (b) 2,3 (c) 6,3 (d) 6,-32 | 11 |

588 | An example of the even number of terms in an arithmetic progression written in a symmetrical format is? A. ( a-d, a+d, a+2 d, a+4 d ) B. ( a-2 d, a+d, a+2 d, a+3 d ) c. ( a-3 d, a-d, a+d, a+3 d ) D. ( a-2 d, a-d, a, a+d, a+2 d ) | 11 |

589 | ( boldsymbol{C}_{mathbf{1}}+mathbf{2} boldsymbol{C}_{mathbf{2}}+mathbf{3} boldsymbol{C}_{mathbf{3}}+ldots ldots+boldsymbol{n} boldsymbol{C}_{boldsymbol{n}} ) is equal to A ( cdot 2^{n-1} ) B . ( 2^{n+1} ) c. ( n .2^{n-1} ) D ( cdot n cdot 2^{n+1} ) | 11 |

590 | According to the property of A.P. If each term of a given A.P ( (1,2,3, ldots) ) is divided by a non-zero constant ( t, ) then the resulting sequence will be A. ( _{text {A.P }}left(frac{1}{t^{prime}}, frac{2}{t^{prime}}, frac{3}{t} ldotsright), ) with common difference ( frac{1}{t} ) B. G.P ( left(frac{1}{t}, frac{2}{t}, frac{3}{t} ldotsright) ), with common difference ( frac{1}{t} ) C. ( _{text {H.P }}left(frac{1}{t}, frac{2}{t}, frac{3}{t} cdotsright) ), with common difference ( frac{1}{t} ) D. A.P ( (t, 2 t, 3 t ldots), ) with common difference ( t ) | 11 |

591 | ( boldsymbol{x}=mathbf{1}+boldsymbol{a}+boldsymbol{a}^{2}+ldots infty, boldsymbol{y}=mathbf{1}+boldsymbol{b}+ ) ( boldsymbol{b}^{2}+ldots infty quad ) then ( quad mathbf{1}+boldsymbol{a} boldsymbol{b}+boldsymbol{a}^{2} boldsymbol{b}^{2}+ ) ( ldots infty= ) | 11 |

592 | The sum of 3 numbers in an AP is 111, and the differences of the squares of the greatest and least is ( 1776 . ) The smallest number is A . 37 B. 25 ( c cdot 49 ) D. 32 | 11 |

593 | ( x ) and ( y ) are two ( +v e ) numbers suchs that ( x y=1 . ) Then the minimum value of ( x+y ) is A . 4 B. ( frac{1}{4} ) ( c cdot frac{1}{2} ) D. 2 | 11 |

594 | Find the fifth term of the G.P. ( frac{3}{2}, frac{3}{4}, frac{3}{8} ) | 11 |

595 | Does the following series form an ( boldsymbol{A P} ) ? ( 0.2,0.22,0.222,0.2222, ldots . ) A. Yes B. No c. Ambiguous D. Data insufficient | 11 |

596 | ( ln ) a G.P., ( t_{2}=frac{3}{5} ) and ( t_{3}=frac{1}{5} . ) Then the common ratio is A ( cdot frac{1}{5} ) B. ( frac{1}{3} ) c. 1 D. | 11 |

597 | In a multiplying the previous term by a constant. A. geometric sequence B. arithmetic sequence c. geometric series D. harmonic sequence | 11 |

598 | Find the value of ( x ) such that ( 1+4+ ) ( mathbf{7}+mathbf{1 0}+ldots ldots+boldsymbol{x}=mathbf{7 1 5} ) | 11 |

599 | If a clock strikes once at 1 o’clock, twice at 2 o’clock and so on, how many times will it strike in a day? | 11 |

600 | What is the ( 7^{t h} ) term in an infinite G.P. ( 3+42+588+dots ? ) A . 12588608 B . 22588608 c. 32588608 D. 42588608 | 11 |

601 | ( f_{a, b, c, d} ) are positive real numbers, then show that ( (a+b+c+ ) ( d)left(frac{1}{a}+frac{1}{b}+frac{1}{c}+frac{1}{d}right) geq 16 . ) What happens when the numbers are all equal? | 11 |

602 | Find three numbers in G.P. whose sum is ( 13 & ) sum of those squares is ( 91 ? ) | 11 |

603 | Sum of ( n ) terms: ( 5+9+13+17+ ) ( mathbf{2 1}+mathbf{2 5}+dots ) | 11 |

604 | 1200 – 21UFRS) 3. The sum of the first n terms of the series 12 + 2.22 +32 +2.42 +52 + 2.62 +…….. is when n is even. When n is odd, the sum is (1988 – 2 Marks n (n + | 11 |

605 | Find the ( 10^{t h} ) term of a ( G P ) whose ( 8^{t h} ) term is 192 and the common ratio is 2 | 11 |

606 | ( p^{t h}, q^{t h} ) and ( r^{t h} ) terms of an A.P. are ( a, b, c ) respectively, then show that (i) ( boldsymbol{a}(boldsymbol{q}-boldsymbol{r})+boldsymbol{b}(boldsymbol{r}-boldsymbol{p})+boldsymbol{c}(boldsymbol{p}-boldsymbol{q})=mathbf{0} ) (ii) ( (a-b) r+(b-c) p+(c-1) q=0 ) | 11 |

607 | Find the sum of Arithmetic progressions of 20 terms ( 1,5,9,13, dots . ) Also find last term. | 11 |

608 | If sum to infinity of series ( 3-5 r+ ) ( 7 r^{2}-9 r 3+. ) is ( 14 / 9, ) find ( r ) | 11 |

609 | How will you identify the given sequence is a geometric sequence? A. common difference B. common ratio c. number of terms D. number of constant | 11 |

610 | Column II gives ( n^{t h} ) term for AP given in column I. Match them correctly. A. ( 119,136,153,170 dots ) 1. 13 – ( mathbf{3} n ) B. ( 7,11,15,19, dots . ) 2. ( 9-5 n ) C. ( 4,-1,-6,-11, dots ) ( 3.3+4 n ) D. ( 10,7,4,3 dots . ) ( 4.17 n+ ) ( mathbf{1 0 2} ) A ( cdot A rightarrow 3, B rightarrow 2, C rightarrow 1, D rightarrow 4 ) B. A ( rightarrow 4, ) В ( rightarrow 3, C rightarrow 2, D rightarrow 1 ) ( mathrm{c} cdot mathrm{A} rightarrow 3, mathrm{B} rightarrow 4, mathrm{C} rightarrow 2, mathrm{D} rightarrow 1 ) D. A ( rightarrow 1, ) В ( rightarrow 3, ) С ( rightarrow 4, ) D ( rightarrow 2 ) | 11 |

611 | After the first term in a sequence of positive integers, the ratio of each term to the term immediately preceding it is 2 to ( 1 . ) What is the ratio of the ( 8^{t h} ) term in this sequence to the ( 5^{t h} ) term? ( A cdot 6 ) to 1 B. 8 to 5 ( c cdot 8 ) to 1 D. 64 to 1 E. 256 to 1 | 11 |

612 | If ( 1+a+a^{2}+a^{3}+dots dots+a^{n}=(1+ ) ( a)left(1+a^{2}right)left(1+a^{4}right) ) then ( n ) is given by ( mathbf{A} cdot mathbf{3} ) B. 5 ( c cdot 7 ) D. | 11 |

613 | Find ( 20^{t h} ) term from the end of an A.P ( mathbf{3}, mathbf{7}, mathbf{1 1}, dots dots dots dots mathbf{. 4 0 7} ) | 11 |

614 | Find the sum of the following APs: ( 0.6,1.7,2.8, dots dots ) to 100 terms A. 5475 B. 5505 ( c cdot 6589 ) D. 3844 | 11 |

615 | Let the positive numbers ( a, b, c, d ) be in A.P. Then ( a b c, a b d, a c d, b c d ) are A. Not in A.P./G.P/H.P. B. in A.P. c. in G.P. D. in H.P. | 11 |

616 | What is the ( 5^{t h} ) term of this sequence? ( mathbf{5}, mathbf{1 0}, mathbf{2 0}, dots ) A . 30 B . 40 c. 50 D. 80 | 11 |

617 | Find three numbers a, b, c, between 2 and 18 such that (1) their sum is 25 Go the numbers 2, a, b sare consecutive terms of an A.P. and (iii) the numbers b, c, 18 are consecutive terms of a G.P. | 11 |

618 | What will be the next number in this sequence ( 21 frac{1}{3}, 16,12,9, ? ) ( A cdot 7 ) B. 6 c. 6.75 D. 5 | 11 |

619 | The sum of three numbers in A.P is 27 . Their product is is ( 405 . ) Find the numbers. | 11 |

620 | If the roots of ( a x^{3}+b x^{2}+c x+d=0 ) are in H.P., then the roots of ( d x^{3}-c x^{2}+ ) ( b x-a=0 ) are in A. A.P в. с. c. н. D. A.G.P | 11 |

621 | Calculate the geometric mean of 3 and ( mathbf{2 7} ) ( mathbf{A} cdot mathbf{3} ) B. 6 ( c .9 ) D. 12 | 11 |

622 | Find out which of the following sequences are arithmetic progressions. For those which are arithmetic progressions, find out the common difference. ( 1^{2}, 5^{2}, 7^{2}, 73, dots ) | 11 |

623 | The sum of the nth term of the series ( 1.2 .5+2.3 .6+3.4 .7+ldots . . ) n terms is | 11 |

624 | The Geometric mean of ( 4,4^{2}, 4^{3}, cdots, 4^{n} ) is A. ( 2^{frac{2}{2}} ) ( frac{2}{2} ) B. ( 2^{frac{n-1}{2}} ) c. ( 4^{n+1} ) D. ( 2^{n+1} ) | 11 |

625 | If a, b, c are in G.P., then the equations ax? + 2bx+c = 0 wird e f . and dx² +2ex + f = 0 have a common root if 7 are in — (1985 – 2 Marks) (a) A.P. (b) GP. (c) H.P. (d) none of these cm of the first n terms of the series | 11 |

626 | Find the first term of an A.P. in which sum of any number of terms is always three times the squared number of these terms. | 11 |

627 | In a GP the sum of three numbers is 14 if 1 is added to first two numbers and the third number is decreased by ( 1, ) the series becomes ( A P, ) find the geometric sequence. This question has multiple correct options ( mathbf{A} cdot 2,4,8 ) в. 8,4,2 ( c .6,18,54 ) D. 8,16,32 | 11 |

628 | Which term of the ( A P: 3,8,13,18, . . ) is ( mathbf{7 8} ? ) ( mathbf{A} cdot t_{1} ) B . ( t_{18} ) ( c cdot t_{14} ) D. None of these | 11 |

629 | Which one of the following is a general form of geometric progression? A. 1,1,1,1,1 в. 1,2,3,4,5 c. 2,4,6,8,10 D. -1,2,-3,4,-5 | 11 |

630 | The sum of first 20 terms of the series ( 1,6,13,22- ) is A . 5580 B. 5780 ( c .7789 ) D. 1237 | 11 |

631 | Find the square of common ratio in given G.P. ( sqrt{mathbf{2}}, sqrt{mathbf{6}}, mathbf{3} sqrt{mathbf{2}}, mathbf{3} sqrt{mathbf{6}} ) | 11 |

632 | The 4 th term of a geometric progression is ( frac{2}{3} ) and 7 th term is ( frac{16}{81} . ) Find the Geometric series. A ( cdot frac{9}{4},-frac{3}{2}, 1,-frac{2}{3}, dots dots ) в. ( frac{9}{4}, frac{3}{2}, 1, frac{2}{3}, ldots . . . ) c. ( -frac{9}{4}, frac{3}{2}, 1, frac{2}{3}, ldots ldots ) D. None of these | 11 |

633 | If ( S=frac{1}{1^{2}}+frac{1}{2^{2}}+frac{1}{3^{2}}+frac{1}{4^{2}}+dots dots dots dots ) to ( infty ) then find ( frac{1}{1^{2}}+frac{1}{3^{2}}+frac{1}{5^{2}}+frac{1}{7^{2}}+dots dots dots dots ) ( infty ) in terms of ( S ) | 11 |

634 | Does there exist a geometric progression containing 27,8 and 12 as three of its terms? | 11 |

635 | The sum of the first 6 terms of a G.P. is 9 times the sum of the first 3 terms; find the common ratio. | 11 |

636 | A geometric series consists of four terms and has a positive common ratio. The sum of the first two terms is 8 and the sum of the last two terms is ( 72 . ) Find the series | 11 |

637 | Find the sum to indicated number of terms in each of the geometric progression. Exercise 7 to 10 ( mathbf{0 . 1 5}, mathbf{0 . 0 1 5}, mathbf{0 . 0 0 1 5}, mathbf{2 0} ) terms | 11 |

638 | Evaluate ( mathbf{7}+mathbf{7 7}+mathbf{7 7 7}+ldots ldots ldots ldots . . ) upto ( n ) terms. A ( cdot frac{7}{81}left[10^{n+1}-9 nright] ) B. ( frac{7}{81}left[10^{n+1}-9 n-10right] ) c. ( frac{7}{81}left[10^{n}-9 n-10right] ) D. ( frac{7}{81}left[10^{n+1}-n-10right] ) | 11 |

639 | Find out which of the following sequences are arithmetic progressions. For those which are arithmetic progressions, find out the common difference. ( mathbf{0},-mathbf{4},-mathbf{8},-mathbf{1 2}, dots ) | 11 |

640 | Is 184 a term of the sequence ( 3,7,11, ldots ) ( ? ) | 11 |

641 | Which term of the ( A P 4,9,14,19, dots ) is ( mathbf{1 0 9} ? ) ( mathbf{A} cdot 14^{t h} ) B. ( 18^{text {th }} ) ( c cdot 22^{n d} ) D. ( 16^{text {th }} ) | 11 |

642 | 890 f(0), then the value 7. Letf(0) = , and S= 1+(cot )* of /25 – 8 is 0=1° | 11 |

643 | Find the mean of 2 and 8 | 11 |

644 | What is the arithmetic mean of the progression ( 11,22,33,44,55,66,77 ? ) A . 44 в. 208 c. 308 D. 48 | 11 |

645 | Let the sequences ( a_{1}, a_{2}, a_{3}, dots dots, a_{n}, dots ) from an AP. Then ( a_{1}^{2}-a_{2}^{2}+a_{3}^{2}-a_{4}^{2}+ ) ( ldots+a_{2 n-1}^{2}-a_{2 n}^{2} ) is equal to A ( cdot frac{n}{2 n-1}left(a_{1}^{2}-a_{2 n}^{2}right) ) B. ( frac{2 n}{n-1}left(a_{2 n}^{2}-a_{1}^{2}right) ) c. ( frac{n}{n+1}left(a_{1}^{2}+a_{2 n}^{2}right) ) D. None of these | 11 |

646 | f ( a, b, c, d ) are in a ( G . P ., ) then show that ( (a-d)^{2}=(b-c)^{2}+ ) ( (c-a)^{2}+(d-b)^{2} ) | 11 |

647 | Write first four terms of the AP, when the first term ( a ) and the common difference ( d ) are given as follows ( a= ) ( -1.25, d=-0.25 ) A. First four terms of the given AP are -1.25,-1.70,-1.95,-2.00 B. First four terms of the given AP are -1.25,-1.50,-1.75,-2.00 c. Not an AP D. None of these | 11 |

648 | The 17 th term of an AP exceeds its 10 th term by 7. Find the common difference. | 11 |

649 | Find the ( 15^{t h} ) term of the series ( mathbf{3}, mathbf{9}, mathbf{1 5}, mathbf{2 1}, mathbf{2 7}, mathbf{3 3}, dots ) | 11 |

650 | Find the arithmetic mean of the series: ( mathbf{1}, mathbf{3}, mathbf{5} dots dots dots dots(mathbf{2 n}-mathbf{1}) ) ( A cdot n ) B. ( 2 n ) ( c cdot frac{n}{2} ) D. ( n-1 ) | 11 |

651 | If an A.P is given by ( 17,14, ldots ldots-40 ) then 6 th term from the end is A ( .-22 ) B. -28 c. -25 D. 26 | 11 |

652 | The sum of two numbers is 6 times their geometric mean show that numbers are in the ratio ( (3+2 sqrt{2}): ) ( (3-2 sqrt{2}) ) | 11 |

653 | Find the sum of the following series to ( n ) terms: ( 1 times 2+2 times 3+3 times 4+4 times 5+dots ) A ( cdot frac{n}{4}(n-1)(n+2) ) B ( cdot frac{n}{3}(n-1)(n-2) ) c. ( frac{n}{2}(n-1)(n+1) ) D ( cdot frac{n}{3}(n+1)(n+2) ) | 11 |

654 | ( ln ) an A.P if sum of its first ( n ) terms is ( 3 n^{2}+5 n ) and its ( k^{t h} ) term is ( 164, ) find the value of ( k ) | 11 |

655 | For ( 0<a<x, ) then the minimum value of the function ( log _{a} x+log _{x} a ) is A . B. 2 ( c cdot 4 ) D. | 11 |

656 | The sum of four consecutive numbers in AP is 32 and the ratio of the product of the first and the last to the product of two middle terms is ( 7: 15 . ) Find the numbers. | 11 |

657 | If ( a, b, c, d, e ) are five positive numbers then This question has multiple correct options ( ^{mathbf{A}} cdotleft(frac{a}{b}+frac{b}{c}right)left(frac{c}{d}+frac{d}{e}right) geq 4 cdot sqrt{frac{a}{e}} ) B ( cdotleft(frac{a}{b}+frac{c}{d}right)left(frac{b}{c}+frac{d}{e}right) geq 4 cdot sqrt{frac{a}{e}} ) c. ( frac{a}{b}+frac{b}{c}+frac{c}{d}+frac{d}{e}+frac{e}{a} geq 5 ) D. ( frac{b}{a}+frac{c}{b}+frac{d}{c}+frac{e}{d}+frac{a}{e} geq frac{1}{5} ) | 11 |

658 | The mean of five numbers in AP is 89. The product of first and last terms is ( 7885 . ) The AM of first, third and fifth term is ( mathbf{A} cdot 83 ) B. 86 ( c .89 ) D. 90 | 11 |

659 | The first term of arithmetic progression is 1 and the sum of the first nine terms equal to ( 369 . ) The first and the ninth term of a geometric progression conicide with the first and the ninth term of the arithmetic progression. Find the seventh term of the geometric progression. | 11 |

660 | Find the sum ( sum_{i=1}^{10} 8 *(1 / 4)^{i-1} ) | 11 |

661 | STATEMENT-1: If ( a b^{2} c^{3}, a^{2} b^{3} c^{4}, a^{3} b^{4} c^{5} ) are in A.P. ( (a, b, c>0), ) then the minimum value of ( a+b+c ) is 3 STATEMENT-2: Arithmetic mean of any two numbers is greater than geometric mean of the numbers. A. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is the correct explanation for STATEMENT-1 B. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1 C . STATEMENT-1 is True, STATEMENT-2 is False D. STATEMENT-1 is False, STATEMENT-2 is True | 11 |

662 | The sum of the first ( n ) terms of an A.P is ( 4 n^{2}+2 n . ) Find the ( n ) th term of this A.P. A. ( 5 n-2 ) B. ( 8 n-2 ) c. ( 3 n-2 ) D. None of these | 11 |

663 | Find the sum of the geometric series ( 4+2+1+ldots+frac{1}{16} ) A ( cdot frac{17}{16} ) в. ( frac{107}{16} ) c. ( frac{117}{16} ) D. ( frac{12}{16} ) | 11 |

664 | Find the A.P. if the 6 th term of the A.P.is 19 and the 16 th term is 15 more than the 11 th term. | 11 |

665 | If fourth therm of an ( A . P ) is thrice its first term and seventh term ( -2(text {thirdterm})=1 ) then its common difference is ? A . 1 B . 2 ( c .-2 ) D. | 11 |

666 | Match the entries of :List-A and List-B. | 11 |

667 | Sum of the series ( sum_{r=1}^{88}(-1)^{r+1} frac{1}{sin ^{2}(r+1)^{circ}-sin ^{2} 1^{circ}} ) equal to A ( cdot frac{cot 2^{circ}}{sin 2^{circ}} ) B. ( frac{-cot 2^{circ}}{sin 2^{circ}} ) c. ( cot 2^{circ} ) D. ( frac{cot 2^{circ}}{sin ^{2} 2^{circ}} ) | 11 |

668 | An arithmetic sequence starts as ( mathbf{5}, mathbf{9}, mathbf{1 3}, dots dots dots, . ) What is the next term? Is 2012 a term of this sequence? Why? | 11 |

669 | Find out which of the following sequences are arithmetic progressions. For those which are arithmetic progressions, find out the common difference. ( -225,-425,-625,-825, dots ) | 11 |

670 | An Ap consists of 50 terms of which 3 rd term is 12 and the last term is ( 106 . ) Find the 29 th term. | 11 |

671 | Find the sum of odd integers from 1 to ( mathbf{2 0 0 1} ) | 11 |

672 | ( frac{frac{1}{1.2 .3}+frac{1}{2.3 .4}+frac{1}{3.4 .5}+ldots .+}{m(n+1)(n+2)}=frac{n(n+3)}{4(n+1)(n+2)} ) | 11 |

673 | For a set of positive numbers, consider the following statements: 1. If each number is reduced by ( 2, ) then the geometric mean of the set may not always exists. 2. If each number is increased by 2 , then the geometric mean of the set is increased by 2 Which of the above statements is/are correct? A. 1 only B. 2 only c. Both 1 and 2 D. Neither 1 nor 2 | 11 |

674 | The sum of integers from 1 to 100 which are divisible by 2 or 5 is- A. 300 B. 3050 c. 3200 D. 3250 | 11 |

675 | Find the ( 21^{s t} ) term of an A.P. whose ( 1^{s t} ) term is 8 and the ( 15^{t h} ) term is 120 A . 167 в. 165 c. 168 D . 169 | 11 |

676 | If the ( 10^{t h} ) term of a G.P. is 9 and ( 4^{t h} ) term is ( 4, ) then its ( 7^{t h} ) term is A. 6 B. 14 c. ( frac{27}{14} ) D. ( frac{56}{15} ) | 11 |

677 | The geometric mean if the series ( 1,2,4, dots .2^{n} ) is ( mathbf{A} cdot 2^{n+(1 / 2)} ) B. ( 2^{(n+1) / 2} ) ( mathrm{c} cdot 2^{n-(1 / 2)} ) ( D cdot 2^{n / 2} ) | 11 |

678 | Find the ( 20^{t h} ) term and ( n^{t h} ) term of the following G.P.5, 25, 125,…. | 11 |

679 | Find the value of ( 13+21+29+. .173 ) | 11 |

680 | If ( boldsymbol{alpha}=mathbf{1} / mathbf{4} ) and ( boldsymbol{P}_{boldsymbol{n}} ) denotes the perimeter of the nth square then ( sum_{n=1}^{infty} P_{n} ) equals A ( cdot frac{8}{3}(4+sqrt{10}) ) B. ( frac{8}{3} ) ( c cdot frac{16}{3} ) D. None of these | 11 |

681 | If ( log _{10} a, log _{10}, log _{10} c ) are in A.P., then ( a, b, c ) must be in A. A.P. в. G.P. c. н.P. D. None of these | 11 |

682 | ( sum_{r=1}^{50}left[frac{1}{49+r}-frac{1}{2 r(2 r-1)}right]= ) A ( cdot frac{1}{50} ) в. ( frac{1}{99} ) c. ( frac{1}{100} ) D. ( frac{1}{101} ) | 11 |

683 | In a G.P. the ( 3^{r d} ) term is ( 24 & 6^{t h} ) term is 192. Find the ( 10^{t h} ) term. | 11 |

684 | For given G.P. ( 1, frac{1}{10}, frac{1}{100}, frac{1}{1000}, frac{1}{10000}, . . ) find the common ratio. A ( cdot frac{1}{10} ) B. ( frac{1}{100} ) c. ( frac{1}{50} ) D. ( frac{3}{10} ) | 11 |

685 | The ( n^{t h} ) term of a GP is given by A ( cdot t_{n}=a r^{n+1} ) В . ( t_{n}=a r^{n-1} ) C ( cdot t_{n}=a r^{n div 1} ) D. ( t_{n}=a r^{2 n-1} ) | 11 |

686 | Find four numbers forming a geometric progression in which the sum of the extreme terms is 112 and the sum of the middle terms is 48 . | 11 |

687 | Find Sum of Series ( tan ^{2} x tan 2 x+ ) ( frac{1}{2} tan ^{2} 2 x tan 4 x+frac{1}{2^{2}} 4 x tan 8 x+dots ) ( n ) terms. | 11 |

688 | The airthmatic mean of ( 1+sqrt{2} ) and ( mathbf{7}+mathbf{5} sqrt{mathbf{2}} ) is ( sqrt{boldsymbol{a}}+sqrt{boldsymbol{b}} . ) Then ( mathbf{a}-boldsymbol{b}= ) ( A cdot-1 ) в. ( c cdot 2 ) ( D cdot-2 ) | 11 |

689 | If a polynomial ( f(x)=4 x^{4}-a x^{3}+ ) ( b x^{2}-c x+5,(a, b, c in R) ) has four positive real zeros ( r_{1}, r_{2}, r_{3}, r_{4} ) such that ( frac{r_{1}}{2}+frac{r_{2}}{4}+frac{r_{3}}{5}+frac{r_{4}}{8}=1, ) then value of ( a ) is A . 20 B. 21 c. 19 D. 22 | 11 |

690 | If ( a=1 ) and ( r=frac{2}{3}, ) find ( (a) T_{n} ) (b) ( T_{4} ) | 11 |

691 | Last digit in ( 2^{2^{n}}+1, n in N, n neq 1 ) is This question has multiple correct options ( A cdot 7 ) B. 3 ( c .5 ) D. | 11 |

692 | Which of the following option will complete the given series ( mathbf{1}, mathbf{6}, mathbf{1 5}, ?, mathbf{4 5}, mathbf{6 6}, mathbf{9 1} ? ) A . 25 B . 26 c. 27 D . 28 | 11 |

693 | An organisation plans to plant saplings in 25 streets in a town in such a way that one sapling for the first street, two for the second, four for the third, eight for the fourth street and so on. How many saplings are needed to complete the work? | 11 |

694 | The 17 th term of an ( A . P . ) is 5 more than twice its ( 8 t h ) term. If the ( 11 t h ) term of the ( A . P ) is 43 , find the ( n t h ) term. | 11 |

695 | ( fleft(a_{1}, a_{2}, dots a_{n} ) be an A.P.of tive terms, right. then ( sum_{k=1}^{n} a_{k} geq n sqrt{a_{1}^{2}+(n-1) d a_{1}} ) where dis common difference of A.P. If you think this is true write 1 otherwise write 0 ? | 11 |

696 | If a>0, b>0 and c>0, prove that 1 1 1 (a +b+c)(4++129 -+ b + – + (a a b | 11 |

697 | ( 3,5,7,9,11,13,15 dots ) is an A. Geometric progression B. Arithmetic series C. Arithmetic progression D. Harmonic progression | 11 |

698 | Find the value of ( mathrm{K} ) so that ( (boldsymbol{K}+ ) 2) ( ,(4 K-6) ) and ( (3 K-6) ) are three continuous terms of AP. | 11 |

699 | Evaluate the following: ( sum_{n=1}^{11}left(2+3^{n}right) ) | 11 |

700 | The value of ( sum_{i=1}^{n} sum_{j=1}^{i} sum_{k=1}^{j} 1= ) | 11 |

701 | If the sum of the first ten terms of the series 16 + 3 , + 4 m 4- + 15 …, 5 5) [JEE M 2016] then m is equal to : (a) 100 (c) 102 (b) 99 (d) 101 C C | 11 |

702 | If ( -frac{pi}{2}<theta<frac{pi}{2} ), then the minimum value of ( cos ^{3} theta+sec ^{3} theta ) is ( A ) B. 2 ( c cdot 0 ) D. none of these | 11 |

703 | If ( 1+sin x+sin ^{2} x+sin ^{3} x+ldots infty ) is equal to ( 4+2 sqrt{3}, 0<x<pi, ) then ( x ) is equal to ( ^{A} cdot frac{pi}{6} ) в. c. ( frac{pi}{3} ) or ( frac{pi}{6} ) D. ( frac{pi}{3} ) or ( frac{2 pi}{3} ) | 11 |

704 | Sum of the series, ( mathbf{1},-boldsymbol{a}, boldsymbol{a}^{2},-boldsymbol{a}^{3}, ldots quad boldsymbol{n} quad ) terms ( quad(boldsymbol{i} boldsymbol{f} quad boldsymbol{a} ) 土1)。 | 11 |

705 | Find the sum to indicated number of terms in the geometric progression given: ( mathbf{0 . 1 5}, mathbf{0 . 0 1 5}, mathbf{0 . 0 0 1 5}, ldots . . mathbf{2 0} ) terms | 11 |

706 | Find first term ‘a’ and common difference ‘d’ for the following AP. ( sqrt{2}, sqrt{8}, sqrt{18}, sqrt{32}, dots ) B. ( a=sqrt{2}, d=sqrt{2} ) c. Its not an AP D. None of these | 11 |

707 | Sum of the series, ( x^{3}, x^{5}, x^{7}, ldots quad n quad ) terms ( (i f quad x neq ) 土1)。 | 11 |

708 | If ( s_{n}=sum_{n=1}^{n} frac{1+2+2^{2}+ldots t o n t e r m s}{2^{n}} ) then ( s_{n} ) is equal to A ( cdot 2^{n}-(n+1) ) B. ( 1-frac{1}{2^{n}} ) c. ( _{n-1}+frac{1}{2^{n}} ) D. ( 2^{n}-1 ) | 11 |

709 | The sequence formed when every term of an arithmetic progression is multiplied or divided by a fixed number is A. A geometric progression B. An arithmetic progression C. Not an arithmetic progression D. A list of numbers with no pattern | 11 |

710 | The mean marks got by 300 students in the subject of statistics was ( 45 . ) The mean of the top 100 of them was found to be 70 and the mean of the last 100 was known to be 20 , then the mean of the remaining 100 students is A . 45 B. 58 c. 68 D. 88 | 11 |

711 | In the series ( 20,18,16, ldots ldots ldots,-2 ) is the term ( mathbf{A} cdot 10^{t h} ) B. ( 11^{text {th }} ) ( mathbf{c} cdot 12^{t h} ) D. ( 13^{text {th }} ) | 11 |

712 | Assertion(A): The minimum value of ( 16 cot x+9 tan x ) is 3 Reason (R): For two positve real numbers a and b, ( boldsymbol{A} . boldsymbol{M} geq boldsymbol{G} . boldsymbol{M} ) This question has multiple correct options A. Both A and R are true and ( R ) is the correct explanation of A B. Both A and R are true and R is not the correct explanation of ( mathrm{c} . ) A is true and ( mathrm{R} ) is false D. A is false and R is true | 11 |

713 | ( frac{1}{c},left(frac{1}{c a}right)^{frac{1}{2}}, frac{1}{a} ) is in ( A cdot A P ) в. GP ( c . ) не D. NONE | 11 |

714 | Identify if a given sequence of numbers is an arithmetic progression or not. a) ( 8,17,26,35, dots ) b) ( -4,-9,-16,-25, ) | 11 |

715 | If ( a_{1}, a_{2}, a_{3}, a_{4}, a_{5} ) are in A.P.with common difference ( neq 0, ) then find the value of ( sum_{i=1}^{5} a_{i} ) when ( a_{3}=2 ) | 11 |

716 | If between any two quantities there be inserted two arithmetic means ( boldsymbol{A}_{1}, boldsymbol{A}_{2} ) two geometric means ( G_{1}, G_{2}: H_{1} H_{2}= ) ( boldsymbol{A}_{mathbf{1}}+boldsymbol{A}_{mathbf{2}}: boldsymbol{H}_{mathbf{1}}+boldsymbol{H}_{mathbf{2}} ) | 11 |

717 | What is the sum of all even numbers between ( 500 & 600 ? ) ( mathbf{A} cdot 26950 ) B. 27500 c. 27950 D. 26500 | 11 |

718 | The A.M. of the series ( 1,2,4,8,16, dots . ., 2 ) is A ( cdot frac{2^{n}-1}{n} ) B. ( frac{2^{n+1}-1}{n+1} ) c. ( frac{2^{n}-1}{n+1} ) ( frac{2^{n+1}-1}{n} ) | 11 |

719 | Sum ( : frac{3}{1^{2} cdot 2^{2}}+frac{5}{2^{2} cdot 3^{2}}+frac{7}{3^{2} cdot 4^{2}}+dots dots ) terms | 11 |

720 | If the arithmetic mean of ( 6,8,5,7, x ) and 4 is ( 7, ) then ( x ) is A . 12 B. 6 ( c cdot 8 ) D. | 11 |

721 | If the sides of a triangle are in A.P, the perimeter of the triangle is ( 30 mathrm{cm} . ) The difference between the longer and shorter side is ( 4 mathrm{cm} . ) Then find the all sides of the triangles. | 11 |

722 | Find ( boldsymbol{K} ) If ( boldsymbol{K}^{2}+boldsymbol{4} boldsymbol{K}+mathbf{8}, boldsymbol{2} boldsymbol{K}^{2}+mathbf{3} boldsymbol{K}+boldsymbol{6} ) and ( 3 K^{2}+4 K+4 ) are any 3 consecutive terms of ( boldsymbol{A} cdot boldsymbol{P} ) | 11 |

723 | If ( a b=2 a+3 b, a>0, b>0 ) then the minimum value of ( a b ) is A . 12 B . 24 ( c cdot frac{1}{4} ) D. none of these | 11 |

724 | Find the sum of all 2 – digit numbers divisible by 3 A. 1335 в. 1445 ( c .1665 ) D. 1555 55 | 11 |

725 | What is the ( 1025^{t h} ) term of the sequence ( mathbf{1}, mathbf{2}, mathbf{2}, mathbf{4}, mathbf{4}, mathbf{4}, mathbf{4}, mathbf{8}, mathbf{8}, mathbf{8}, mathbf{8}, mathbf{8}, mathbf{8}, mathbf{8}, mathbf{8}, mathbf{?} ) | 11 |

726 | A.M. and H.M. between two quantities are 27 and 12 respectively, find their G.M | 11 |

727 | If the sum of 7 terms is ( 49, ) and the sum of 17 terms is ( 289, ) find the sum of ( n ) terms. | 11 |

728 | Find the difference of the arithmetic progression an if ( a_{1}=7 ) and ( a_{3}=16 ) A . 4.5 B. 5 ( c .5 .5 ) D. 6 | 11 |

729 | The sum of 10 terms of GP ( frac{1}{2}+frac{1}{4}+frac{1}{8}+ ) . is- A ( cdot frac{2^{10}-1}{2^{10}} ) B. ( frac{2^{9}-1}{2^{9}} ) c. ( frac{2^{10}-1}{2^{9}} ) D. ( frac{2^{9}-1}{2^{10}} ) | 11 |

730 | The sum of the series ( 1+frac{1.3}{6}+ ) ( frac{1.3 .5}{6.8}+ldots . infty ) is? | 11 |

731 | Find three numbers which form a geometric progression, if their product is 64 and the arithmetic mean is ( frac{14}{3} ) | 11 |

732 | If ( boldsymbol{x} in boldsymbol{R}, ) find the minimum value of the expression ( 3^{x}+3^{1-x} ) begin{tabular}{l} A ( 3 sqrt{2} ) \ hline end{tabular} B. ( 2 sqrt{3} ) ( c cdot 3 sqrt{3} ) D. none of these | 11 |

733 | If three positive real numbers ( a, b, c ) are in ( A . P . ) such that ( a b c=4, ) then the minimum possible value of ( b ) is ( mathbf{A} cdot 2^{3 / 2} ) B. ( 2^{2 / 3} ) ( mathbf{c} cdot 2^{1 / 3} ) D. ( 2^{5 / 2} ) | 11 |

734 | If an ( A . P ) consists of ( n ) terms with first term ( a ) and ( n^{t h} ) term ( l ) show that the sum of the ( m^{t h} ) term from the beginning and the ( m^{t h} ) term from the ends is ( (a+ ) ( l ) | 11 |

735 | If a variate takes values ( a, a r, a r^{2}, ldots ldots a r^{n-1} ) which of the relation between means hold? A ( . A H=G^{2} ) в. ( frac{A+H}{2}=G ) c. ( A>G>H ) D. ( A=G=H ) | 11 |

736 | Which of the following is a geometric series? A ( cdot 2,4,6,8, dots dots ) B ( cdot 1 / 2,1,2,4 ldots ldots ) c. ( 1 / 4,1 / 6,1 / 8,1 / 10, ldots ldots ) D・ ( 3,9,18,36, ldots ) | 11 |

737 | f ( a, b, c, d ) are in G.P., show that ( mathbf{i}) boldsymbol{a}^{2}+boldsymbol{b}^{2}, boldsymbol{b}^{2}+boldsymbol{c}^{2}, boldsymbol{c}^{2}+boldsymbol{d}^{2} ) are in G.P | 11 |

738 | If the arithmetic mean of ( a ) and ( b ) is double of their geometric mean, with ( a>b>0, ) then a possible value for the ratio ( frac{a}{b}, ) to the nearest integer, is A. 5 B. 8 ( c cdot 11 ) D. 14 | 11 |

739 | If the ( 5 t h ) and ( 8 t h ) term of an ( A P ) are 6 and 15 respectively, find the 19 th term A . 48 B . 54 c. 19 D. 57 | 11 |

740 | The ( p^{t h} ) term of an arithmetic progression is ( boldsymbol{q}, ) and the ( boldsymbol{q}^{t h} ) term is ( boldsymbol{p} ) then the ( m^{t h} ) term is: A. ( p+q-m ) в. ( p-q-m ) c. ( p+q+m ) D. None of these | 11 |

741 | The least value of ( 9 sec ^{2} x+16 operatorname{cosec}^{2} x ) is: A . 25 B. 49 c. 81 D. 64 | 11 |

742 | In an arithmetic series ( S_{20}=-850 ) and ( t_{20}=-90, ) find ( t_{1} ) A. 5 B. 4 ( c cdot 3 ) ( D cdot 2 ) | 11 |

743 | If the sides ( a, b, c ) of ( triangle A B C ) are in ( A . P . ) prove that ( a cos ^{2} frac{C}{2}+c cos ^{2} frac{A}{2}=frac{3 b}{2} ) | 11 |

744 | Find the sum of an infinite ( mathrm{G.P}: 1+ ) ( frac{1}{3}+frac{1}{9}+frac{1}{27}+dots dots ) A ( cdot frac{3}{5} ) B. ( frac{3}{2} ) c. ( frac{49}{27} ) D. 8 ( overline{5} ) | 11 |

745 | 16. In the quadratic equation ax2 + bx+c=0, A =b2 – 4ac and a+b, a2 + B2, a3 + B3, are in GP, where a, ß are the root of ax2 + bx+c=0, then (2005S) (a) A+ (b) bA=0 (c) cA=0 (d) A=0 | 11 |

746 | If the ( p^{t h} ) term of an ( A P ) is ( q ) and its ( q^{t h} ) term is ( p, ) then its ( (p+q)^{t h} ) term is ( A cdot p ) B. ( q ) c. ( (p+q) ) D. | 11 |

747 | f ( sum_{r=1}^{n} t_{r}=frac{1}{12} n(n+1)(n+2), ) then the value of ( sum_{r=1}^{n} frac{1}{t_{r}} ) is A ( cdot frac{2 n}{n+1} ) в. ( frac{n}{(n+1)} ) c. ( frac{4 n}{n+1} ) D. ( frac{3 n}{n+1} ) | 11 |

748 | ( ln ) an ( A . P ., ) sum of ( p ) terms ( = ) sum of q terms, then sum of ( (boldsymbol{p}+boldsymbol{q}) ) terms equal to? A . 0 B. 2 c. ( -p q ) ( mathbf{D} cdot-(p+q) ) | 11 |

749 | The sum of ( n ) terms of the series ( 1^{2}+ ) ( mathbf{2} . mathbf{2}^{mathbf{2}}+mathbf{3}^{mathbf{2}}+mathbf{2 . 4}^{mathbf{2}}+mathbf{5}^{mathbf{2}}+mathbf{2 . 6}^{mathbf{2}}+ldots . ) is ( frac{n(n+1)^{2}}{2}, ) where ( n ) is even. Find the sum, where ( n ) is odd | 11 |

750 | The sum of first ( n ) terms of a sequence is ( frac{6^{n}-5^{n}}{5^{n}} ) Find its ( n^{t h} ) term and examine whether the sequence is an A.P of G.P. | 11 |

751 | Is 184 a term of the sequence ( 3,7,11, ldots ) ( ? ) | 11 |

752 | Which term of the sequence ( 8-6 i, 7- ) ( mathbf{4} i, mathbf{6}-mathbf{2} i, ldots ) is purely real ( ? ) A . 10 B. 12 ( c .9 ) D. | 11 |

753 | If ( a>0, b>0, c>0 ) and the minimum value of ( boldsymbol{a}left(boldsymbol{b}^{2}+boldsymbol{c}^{2}right)+boldsymbol{b}left(boldsymbol{c}^{2}+boldsymbol{a}^{2}right)+boldsymbol{c}left(boldsymbol{a}^{2}+boldsymbol{b}^{2}right) ) is ( lambda a b c ), then ( lambda ) is ( A cdot 2 ) в. ( c cdot 6 ) D. | 11 |

754 | Which term of the AP.20, ( 17,14, ldots ldots . ) is the first negative term? A. 8th B. 6th c. 9 th D. 7th | 11 |

755 | The first term of an arithmetic progression is 15 and the last term is ( 85 . ) If the sum of all terms is ( 750, ) what is the 6th term? A . 30 B. 40 ( c cdot 45 ) D. 55 | 11 |

756 | The 4 th term of an A.P. is zero. Prove that the 25 th term of the A.P. is three times its 11 th term. | 11 |

757 | In a multiplying the previous term by a constant. A . arithmetic sequence B. geometric series c. arithmetic series D. harmonic progression | 11 |

758 | How many terms of G.P ( 3,3^{2}, 3^{3}, ldots ) are needed to give the sum ( 120 ? ) | 11 |

759 | – + – + – +….- For a positive integer n, let 1 1 1 1 . a (n)=1+2 3 4 (2″) -1° (a) a(100) 100 (c) a (200) 100 (d) a (200) > 100 | 11 |

760 | Let ‘n’ Arithmetic Means and ‘n’ Harmonic Means are inserted between two positive number’a’ and ‘b’. If sum of all Arithmetic Means is equal to sum of reciprocal all Harmonic means, then product of number is ( A ) B. 2 ( c cdot frac{1}{2} ) D. 3 | 11 |

761 | Find the sum of the first (i) 75 positive integers (ii) 125 natural numbers | 11 |

762 | What is series? A. adding all the numbers B. subtracting all the numbers C. multiplying all the numbers D. dividing all the numbers | 11 |

763 | la, b, c are is G.P and ( a-b, c-a ) and ( b-a ) are in H.P, then ( a+4 b+c ) is equal to | 11 |

764 | ( ln ) a ( G . P ) if ( T_{p-1}+T_{p+1}=3 T_{p} ) then prove that the common ratio of ( G . P ) is an irrational number. | 11 |

765 | Given ( a_{12}=37, d=3, ) find a and ( S_{12} ) ( mathbf{A} cdot 4,246 ) B. 6,268 ( mathbf{c} cdot 9,296 ) D. 3,264 | 11 |

766 | Find the arithmetic mean of 4 and 6 | 11 |

767 | In the sequence ( mathbf{1}, mathbf{2}, mathbf{2}, mathbf{3}, mathbf{3}, mathbf{3}, mathbf{4}, mathbf{4}, mathbf{4}, mathbf{4}, dots, ) where ( boldsymbol{n} ) consecutive terms have the value ( n ), the ( 150^{t h} ) term is A . 17 B . 16 c. 18 D. none of these | 11 |

768 | If the ratio of the sums of first of two A.P.’s is ( (7 n+1):(4 n+27), ) find the ratio of their ( m^{t h} ) terms. | 11 |

769 | Assertion If ( A, B, C ) are acute positive angles then ( frac{(sin A+sin B)(sin B+sin C)(sin C+s)}{sin A sin B sin C} ) 8 Reason ( boldsymbol{A M} geq boldsymbol{G} boldsymbol{M} ) A. Statement-1 is false, statement-2 is true B. Statement-1 is true, statement-2 is true,statement-2 is correct explanation for statement- c. statement-1 is true, statement-2 is true,statement-2 is not a correct explanation for statement- D. Statement-1 is true, statement-2 is false | 11 |

770 | Which one of the following is a series? A. ( 1 times 2 times 3 times 4 times 5 ) в. (0,1,2,3,4) c. {1,2,3,5,6} D. ( sum_{n=1}^{10} n k^{2} ) | 11 |

771 | Let ( r^{t h} ) term of a series is given by, ( boldsymbol{T}_{boldsymbol{r}}= ) ( frac{r}{1-3 r^{2}+r^{4}} ) Then ( lim _{n rightarrow infty} sum_{r=1}^{n} T_{r} ) is A ( cdot frac{3}{2} ) в. ( frac{1}{2} ) ( c cdot frac{-1}{2} ) D. ( frac{-3}{2} ) | 11 |

772 | Sum to ( n ) terms the series: ( mathbf{1} times mathbf{2}+mathbf{2} times mathbf{3}+mathbf{3} times mathbf{4}+mathbf{4} times mathbf{5}+dots ) | 11 |

773 | Find the ( 11 t h ) term from the last term (towards the first term) of the ( boldsymbol{A P} ) : ( mathbf{1 0}, mathbf{7}, mathbf{4}, dots,-mathbf{6 2} ) | 11 |

774 | Find the sum of first eight multiples of three. | 11 |

775 | Find the sum of the series. ( mathbf{1 1}^{mathbf{3}}+mathbf{1 2}^{mathbf{3}}+mathbf{1 3}^{mathbf{3}}+ldots+mathbf{2 8}^{mathbf{3}} ) | 11 |

776 | In an A.P the first term is 2 and the sum of the first five terms is one fourth of the next five terms. Show that 20 th term is -112 | 11 |

777 | State whether the following sequence is an A.P. or not: ( mathbf{3}, mathbf{6}, mathbf{1 2}, mathbf{2 4}, dots ) | 11 |

778 | Write the first term ( a ) and the common difference ( d ) of the ( A P:-5,-1,3,7 dots . . . ) A ( . a=-5, d=4 ) В. ( a=-5, d=-6 ) c. ( a=-5, d=-4 ) D. ( a=5, d=-4 ) 4 | 11 |

779 | 22. A man saves 200 in each of the first three months of his service. In each of the subsequent months his saving increases by 40 more than the saving of immediately previous month. His total saving from the start of service will be 11040 after [2011] (a19 months (b) 20 months (c) 21 months (d) 18 months | 11 |

780 | If the product of two positive numbers is 256 then the least value of their sum is A . 32 B . 16 c. 48 D. 40 | 11 |

781 | The ratio between the sum of ( n ) terms of two A.P.’s is ( 3 n+8: 7 n+15 ).Find the ratio between their 12 th terms. A ( cdot frac{7}{12} ) в. ( frac{7}{16} ) c. ( frac{7}{20} ) D. ( frac{7}{18} ) | 11 |

782 | How many terms of the series ( 54,51,48, dots dots ) be taken so that their sum is ( 513 ? ) A . 18 B . 19 c. 20 D. ( A ) and ( B ) | 11 |

783 | Find the sum of ( 1^{2}-2^{2}+3^{2}-4^{2}+ ) ( +19^{2}-20^{2}=dots dots ) ( mathbf{A} cdot-210 ) в. 120 ( c cdot 210 ) D. -120 | 11 |

784 | Find the second term and ( n ) th term of an AP whose 6 th term is 12 and 8 th term is 22 A ( cdot a_{2}=-9, a_{n}=3 n-43 ) в. ( a_{2}=7, a_{n}=3 n-52 ) c. ( a_{2}=-8, a_{n}=5 n-18 ) D ( cdot a_{2}=-9, a_{n}=5 n-38 ) | 11 |

785 | In the series ( (k b j s s a f e e n o p q f j ) i o ( mathrm{m} mathrm{p} ), the number of consonant present are : A . 11 B . 13 c. 14 D. 12 | 11 |

786 | Let ( T_{r} ) be the ( r^{t h} ) term of an A.P for ( r= ) ( 1,2,3, dots ) if for some positive integers ( boldsymbol{m}, boldsymbol{n}, ) we have ( boldsymbol{T}_{boldsymbol{m}}=frac{mathbf{1}}{boldsymbol{n}} ) and ( boldsymbol{T}_{boldsymbol{n}}=frac{mathbf{1}}{boldsymbol{m}} ) then ( T_{m n} ) equals A ( cdot frac{1}{m n} ) B. ( frac{1}{m}+frac{1}{n} ) c. 1 D. | 11 |

787 | The sum of three consecutive odd numbers is ( 51 . ) Find the numbers. | 11 |

788 | The ( G . M . ) of the numbers ( mathbf{3}, mathbf{3}^{2}, mathbf{3}^{3}, dots, mathbf{3}^{n} ) is A ( cdot frac{2}{n} ) в. ( frac{n-1}{2} ) c. ( frac{n}{3} ) D. ( frac{n+1}{2} ) | 11 |

789 | Find how many terms of G.P. ( frac{2}{9}-frac{1}{3}+frac{1}{2} ldots ) must be added to get the sum equal to ( frac{55}{72} ? ) | 11 |

790 | ( 4, frac{8}{3}, frac{16}{9}, frac{32}{27} dots ) is a A. arithmetic sequence B. geometric sequence C . geometric series D. harmonic sequence | 11 |

791 | Find the mean weight from the following table weight ( mathbf{2 9} ) ( mathbf{3 0} quad mathbf{3 1} ) ( (k g) ) No.of ( 01 quad 04 quad 03 ) children 20 | 11 |

792 | The sum of 12 terms of an A.P., whose first term is ( 4, ) is ( 256 . ) What is the last term? A . 35 B. 36 6 ( c .37 ) D. None | 11 |

793 | The common difference for even number of terms written in symmetrical manner for an arithmetic progression have a common difference of ? ( A cdot d ) B. ( 2 d ) c. ( 3 d ) D. ( -2 d ) | 11 |

794 | Let the sequence ( a_{1}, a_{2}, ldots-ldots–a_{n} ) form an A.P. and let ( a_{1}=0, ) prove that [ frac{boldsymbol{a}_{boldsymbol{3}}}{boldsymbol{a}_{2}}+frac{boldsymbol{a}_{boldsymbol{4}}}{boldsymbol{a}_{boldsymbol{3}}}+frac{boldsymbol{a}_{boldsymbol{5}}}{boldsymbol{a}_{boldsymbol{4}}}+_{——–}+frac{boldsymbol{a}_{boldsymbol{n}}}{boldsymbol{a}_{boldsymbol{n}-boldsymbol{1}}}- ] ( boldsymbol{a}_{2}left(frac{mathbf{1}}{boldsymbol{a}_{2}}+frac{mathbf{1}}{boldsymbol{a}_{3}}+ldots+frac{mathbf{1}}{boldsymbol{a}_{n-2}}right)=frac{boldsymbol{a}_{boldsymbol{n}-mathbf{1}}}{boldsymbol{a}_{2}}+ ) ( frac{boldsymbol{a}_{2}}{boldsymbol{a}_{n-1}} ) | 11 |

795 | Let the sum of the series ( frac{1}{1^{3}}+ ) ( frac{1+2}{1^{3}+2^{3}}+ldots . .+frac{1+2+ldots .+n}{1^{3}+2^{3}+ldots . .+n^{3}} ) upto ( n ) terms be ( S_{n}, n=1,2,3, dots . . ) Then ( S_{n} ) cannot be greater than This question has multiple correct options A ( cdot frac{1}{2} ) B. c. 2 D. 4 | 11 |

796 | The sum to ( n ) terms of the services ( frac{3}{1^{2}}+ ) ( frac{5}{1^{2}+2^{2}}+frac{7}{1^{2}+2^{2}+3^{2}}+dots dots ) is A ( cdot frac{3 n}{n+1} ) В. ( frac{6 n}{n+1} ) c. ( frac{g n}{n+1} ) D. ( frac{12 n}{n+1} ) | 11 |

797 | ( n ) arithmetic means are inserted between 3 and ( 17 . ) If the ratio of last and the first arithmetic mean is ( 3: 1, ) then the value of ( n ) is ( mathbf{A} cdot mathbf{9} ) B. 6 ( c cdot 7 ) D. | 11 |

798 | The ratio of nth term of two A.P.s is ( (14 n-6):(8 n+23), ) then the ratio of their sum of first ( mathrm{m} ) terms is A ( cdot frac{4 m+4}{7 m+24} ) в. ( frac{7 m+1}{4 m+24} ) c. ( frac{7 m+1}{4 m+27} ) D. ( frac{28 m-20}{16 m+15} ) | 11 |

799 | ( boldsymbol{a}=mathbf{1}, boldsymbol{d}=mathbf{2}, ) find ( boldsymbol{S}_{mathbf{1 0}} ) | 11 |

800 | Find the coordinates that was of the equation ( boldsymbol{x}^{3}-boldsymbol{p} boldsymbol{x}^{2}+boldsymbol{q} boldsymbol{x}-boldsymbol{r}=mathbf{0}, ) may be in A.P. and hence solve the equation ( x^{3}-12 x^{2}+34 x-28=0 ) | 11 |

801 | The population of a town is 1200 in 2010. From 2010 to 2015, the population of the town increased by ( 10 % ) per year. Whats the population of the town in ( 2015 ? ) A .4932 .612 B . 3932.612 c. 2932.612 D. 1932.612 | 11 |

802 | The sum of the squares of three distinct real numbers, which are in ( G . P . ) is ( S^{2} . ) If their sum is ( alpha S, ) show that ( alpha^{2} in ) ( left(frac{1}{3}, 1right) cup(1,3) ) | 11 |

803 | Find the Odd one among : 2,5,11,22,32 47 A . 23 B. 36 c. 27 D. 51 | 11 |

804 | ff ( (x)=log _{x} 1 / 9-log _{3} x^{2}(x>1) ) then ( max f(x) ) is equal to | 11 |

805 | How many terms are there in the sequence ( 4,11, ldots ldots ., 298 ? ) | 11 |

806 | Find the sum to 90 terms of the series ( mathbf{5}+mathbf{5 5}+mathbf{5 5 5}+cdots cdots ) A ( frac{50}{81}left[10^{90}-82right] ) в. ( frac{50}{81}left[10^{90}-83right] ) c. ( frac{50}{81}left[10^{90}-80right] ) D. ( frac{50}{81}left[10^{90}-90right] ) | 11 |

807 | The ( 11^{t h} ) term and the ( 21^{s t} ) term of an A.P. are 16 and 29 respectively, then find the first term and common difference. | 11 |

808 | If the angles of a triangle ( A B C ) are in ( boldsymbol{A} cdot boldsymbol{P} ) ( boldsymbol{a}=mathbf{2}, boldsymbol{c}=mathbf{4}, ) then ( boldsymbol{b}= ) A ( cdot 2 sqrt{3} ) 3 B. ( sqrt{21} ) c. 8 D. 14 | 11 |

809 | ( 10^{t h} ) term of ( mathrm{AP}: 2,7,12, ldots . . ) is A . 35 B. 47 c. 55 D. None of the above | 11 |

810 | Find the common difference of the A.P. and write the next two terms: 1. ( , 2.0,2.2,2.4, dots . ) | 11 |

811 | f ( log _{10} 2, log _{10}left(2^{x}-1right) ) and ( log _{10}left(2^{x}+right. ) 3) be three consecutive terms of an A.P. then ( x=log _{2} 5 ) | 11 |

812 | ( 2+2^{2}+2^{3}+ldots ldots+2^{9}=? ) ( mathbf{A} cdot 2044 ) B. 1022 c. 1056 D. None of these | 11 |

813 | Find the sum of ( p ) terms of the series whose ( n^{t h} ) term is ( frac{n}{a}+b ) | 11 |

814 | If the ratio of HM and GM of two quantities is ( 12: 13, ) then the ratio of the number is ? A . 1: 2 B . 2: 3 c. 3: 4 D. None of these | 11 |

815 | Find the value of ( boldsymbol{y} ) if ( mathbf{1}+mathbf{4}+mathbf{7}+mathbf{1 0}+ ) ( ldots .+y=287 ) | 11 |

816 | Which term of the following sequences ( sqrt{3}, 3,3 sqrt{3}, dots ) is 729 | 11 |

817 | Find the value of ( mathrm{k}, ) If ( x, 2 x+k, 3 x+5 ) are consecutive terms in A.P. | 11 |

818 | Find the seventh term of the G.P.: ( sqrt{3}+ ) ( 1,1, frac{sqrt{3}-1}{2} ) | 11 |

819 | Determine an ( A . P ) whose third term is 9 and when fifth term is subtracted from ( 8 t h ) term we get 6 | 11 |

820 | The first term is 1 in the geometric sequence ( 1,-3,9,-27, ldots . . ) What is the SEVENTH term of the geometric sequence? A. -243 B. -30 c. 81 D. 189 ( E .729 ) | 11 |

821 | If an A.P is given by ( 7,12,17,22, ) then ( n ) th term is A ( .2 n+5 ) B. ( 4 n+3 ) ( c cdot 5 n+2 ) D. ( 3 n+4 ) | 11 |

822 | Find the ( 10 t h ) term from end for the ( A . P .3,6,9,12, dots, 300 ) | 11 |

823 | If ( a_{1}=a_{2}=2, a_{n}=a_{n-1}-1(n>2) ) then ( a_{5} ) is ? ( mathbf{A} cdot mathbf{1} ) B. – – ( c .0 ) D. – | 11 |

824 | Prove ( frac{1}{2}+frac{1}{4}+frac{1}{8}+ldots+frac{1}{2^{n}}=1-frac{1}{2^{n}} ) | 11 |

825 | The mth term of A.P is n and its nth term is m. its pth term is ( A cdot m+n+p ) B. ( m+n-p ) ( c cdot m-n+p ) D. None | 11 |

826 | If the sum of ( n ) terms of an A.P. is ( n A+ ) ( n^{2} B, ) where ( A, B ) are constants, then its common difference will be A. ( A-B ) в. ( A+B ) ( c .2 A ) D. 2 ( B ) | 11 |

827 | Find the ( 1.4 .7+2.5 .8+3.6 .9+ ) ( ldots ldots . n^{t h} ) terms | 11 |

828 | ( mathbf{1 0 0}^{2}-mathbf{9 9}^{mathbf{2}}+mathbf{9 8}^{mathbf{2}}-mathbf{9 7}^{mathbf{2}}+mathbf{9 6}^{mathbf{2}}+mathbf{9 5}^{mathbf{2}}+ ) ( ldots .=mathbf{S} ) find ( boldsymbol{S}=? ) | 11 |

829 | Obtain the sum of the first 56 terms of an A.P whose ( 25^{t h} ) and ( 32^{n d} ) terms are 52 and 148 respectively. | 11 |

830 | The sum n terms of two A Ps are in ratio ( frac{7 n+1}{4 n+27} ) Find the ratio of their 11 th terms A ( cdot frac{148}{111} ) в. ( frac{213}{311} ) c. ( frac{221}{343} ) D. ( frac{114}{157} ) | 11 |

831 | If statement ( boldsymbol{P}(boldsymbol{n}) ) is ( ” boldsymbol{3} boldsymbol{n}+1 ) is even” Then verify that statement ( boldsymbol{P}(mathbf{1}) ) is true but ( boldsymbol{P}(2) ) is not true | 11 |

832 | Sum ( 1+2 a+3 a^{2}+4 a^{3}+dots ) to ( n ) terms. A ( cdot frac{1+left(a^{n}right)}{(a-1)^{2}}-frac{n a^{n}}{1+a} ) в. ( frac{1-2left(a^{n}right)}{(a-1)^{2}}+frac{n a^{n}}{1-2 a} ) c. ( frac{1-left(a^{n}right)}{(a-1)^{2}}-frac{n a^{n}}{1-a} ) D. none of these | 11 |

833 | Find the common difference the following A.P ( 1,4,7,10,13,16, dots ) | 11 |

834 | The sum of the series ( (2)^{2}+2(4)^{2}+3(6)^{2}+ldots . ) upto 10 terms is A . 12100 B. 11300 c. 11200 D. 12300 | 11 |

835 | 6. 13-23 + 33–43. (a) 425 (6) 425 (c) 475 [2002] (d) 475 | 11 |

836 | If ( a, b, c ) are in ( A . P . ) and ( P ) is the ( A . M . ) between ( a ) and ( b, ) and ( q ) is the ( A . M ) between ( b ) and ( c, ) show that ( b ) is the A. ( M . ) between ( p ) and ( q ) | 11 |

837 | Find the A.P whose ( 7^{t h} ) and ( 13^{t h} ) terms are respectively 34 and 64 | 11 |

838 | ( 11^{t h} ) term of the ( A P ) is ( :-3,-frac{1}{2}, 2, dots ) is A . 28 B. 22 c. -38 D. -48 | 11 |

839 | Find the sum of the series ( 1 cdot 2+2 ) ( mathbf{3}+mathbf{3} cdot mathbf{4}+cdots+boldsymbol{n}(boldsymbol{n}+mathbf{1}) ) | 11 |

840 | Find the A.P. whose first term is 4 and common difference is – 3 | 11 |

841 | If ( m ) arithmetic means are inserted between 1 and ( 31, ) so that the ratio of the ( 7^{t h} ) and ( (m-1)^{t h} ) means is 5: 9 then the value of ( m ) is ( mathbf{A} cdot mathbf{9} ) B. 11 c. 13 D. 14 | 11 |

842 | The general form of A.P. is ( a, a+d ) ( mathbf{A} cdot a+2 d ) B. ( a-2 d ) ( mathbf{c} cdot a+d ) ( mathbf{D} cdot a-d ) | 11 |

843 | If the sum of the first p terms of an AP is ( a p^{2}+b p, ) find its common difference. | 11 |

844 | Check whether ( 4^{n} ) can end with 0 or not. | 11 |

845 | ( mathbf{A} ) G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying the odd places. Find the common ration of the G.P. | 11 |

846 | Find the first four terms of an ( A P ) whose first term is ( 3 x+y ) and common Difference is ( boldsymbol{x}-boldsymbol{y} ) | 11 |

847 | In an arithmetic progression, ( a_{7}=9 . ) At what value of its difference is the product ( a_{1} a_{2} a_{7} ) the least? | 11 |

848 | Which of the following are APs? If they form an AP, find the common difference ( d ) and write three more terms. -1.2,-3.2,-5.2,-7.2 | 11 |

849 | If ( a, b, c ) are the lengths of sides of a triangle, then the minimum value of ( frac{a}{b}+frac{b}{c}+frac{c}{a} ) ( A ) B. 2 ( c cdot 5 ) D. | 11 |

850 | In a G.P. the first term is ( 7, ) the last term ( 448, ) and the sum ( 889 ; ) find the common ratio. ( A cdot 2 ) B. 3 ( c cdot 4 ) D. 5 | 11 |

851 | The sum of three numbers which form a geometric progression is 13 and the sum of their squares is ( 91 . ) Find the numbers. | 11 |

852 | A sprinter runs 6 meters in the first second of a certain race and increase her speed by ( 25 mathrm{cm} / mathrm{sec} ). in each succeeding second. (This means that she goes ( 6 mathrm{m} 25 mathrm{cm} ). the second second, ( 6 mathrm{m} 50 mathrm{cm} . ) the third second, and so on.) How far does she go during the eight second? A. ( 8.75 mathrm{m} ) B. 7.75 m ( c .8 .25 mathrm{m} ) D. 9.25 m | 11 |

853 | The sum of an infinitely decreasing geometric progression is equal to 4 and the sum of the cubes of its terms is equal to ( 192 . ) Find the first term and the common ratio of the progression. | 11 |

854 | The ( 31 s t ) term of the ( A P ) whose first two terms are respectively -2 and -7 is ( mathbf{A} cdot-152 ) в. 150 c. 148 D. -148 | 11 |

855 | Sum of the natural number between ( mathbf{1 0 0} ) and ( mathbf{2 0 0} ) whose HCF with ( mathbf{9 1} ) should be more than 1 A. 1121 в. 3210 ( c .3121 ) D. 1520 | 11 |

856 | Calculate the fifth term of the sequence ( a_{n}=2(n-1)^{2}-3 ) A . 19 B . 29 c. 39 D. 49 | 11 |

857 | 20 our numbers are chosen at random (without replacement) from the set {1,2,3,…20). Statement-1: The probability that the chosen numbers when [2010] arranged in some order will form an AP is 25 Statement -2: If the four chosen numbers form an AP, then the set of all possible values of common difference is (+1, +2, +3, +4, +5). (a) Statement – 1 is true, Statement -2 is true ; Statement-2 is not a correct explanation for Statement-1 (b) Statement -1 is true, Statment-2 is false (C) Statement -1 is false, Statment-2 is true. (d) Statement -1 is true, Statement -2 is true; Statement-2 is a correct explanation for Statement-1. | 11 |

858 | If the mean of ( 20,14,16,19, p ) and 21 is 27 then find the value of ( ^{prime} p^{prime} ) | 11 |

859 | Sequences and Their limits. An infinitely Decreasing Geometric Progression. limits of Functions. The sum of the terms of an infinitely decreasing geometric progression is equal to the greatest value of the function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}+boldsymbol{3} boldsymbol{x}-boldsymbol{9} ) on the interval [-2,3]( ; ) the difference between the first and the second terms of the progression is ( boldsymbol{f}^{prime}(mathbf{0}) . ) Find the common ratio of the progression. | 11 |

860 | Prove that no matter what the real numbers ( a ) and ( b ) are, the sequence with ( n ) th term ( a+n b ) is always an A.P. What is the common difference? | 11 |

861 | What is the common difference of an A.P. in which ( a_{21}-a_{7}=84 ? ) | 11 |

862 | If ( a_{1}, a_{2}, dots, a_{n} ) are positive real numbers whose product is a fixed number ( c ), the minimum value of ( boldsymbol{a}_{mathbf{1}}+boldsymbol{a}_{mathbf{2}}+ldots+ ) ( boldsymbol{a}_{boldsymbol{n}-mathbf{1}}+boldsymbol{2} boldsymbol{a}_{boldsymbol{n}} ) is A ( cdot n(2 c)^{1 / n} ) B cdot ( (n+1) c^{1 / n} ) ( mathbf{c} cdot 2 n c^{1 / n} ) D・ ( (n+1)(2 c)^{1 / n} ) | 11 |

863 | The fourth term of a G.P. is 27 and the 7th term is ( 729, ) find the 5 th term of G.P. | 11 |

864 | Identify whether the following sequence is a geometric sequence or not. ( frac{1}{2}, frac{2}{4}, frac{4}{8}, frac{8}{16} ) | 11 |

865 | Calculate the sum of first 20 terms of the G.P. ( -1,1,-1,1 dots ) ( mathbf{A} cdot mathbf{0} ) B. ( c cdot 2 ) D. 3 | 11 |

866 | If one ( G . M .^{prime} G^{prime} ) and two arithmetic means ( p ) and ( q ) be inserted between any two given number then ( G^{2} ) is: ( mathbf{A} cdot(2 p+q)(2 q-p) ) в. ( (2 p-q)(q+p) ) c. ( (2 p-q)(2 q-p) ) D. ( (p+q)(q+p) ) | 11 |

867 | Let ( left{a_{n}right} ) and ( left{b_{n}right} ) are two sequences given by ( a_{n}=(x)^{1 / 2^{n}}+ ) ( (boldsymbol{y})^{1 / 2^{n}} ) and ( boldsymbol{b}_{n}=(boldsymbol{x})^{1 / 2^{n}}-(boldsymbol{y})^{1 / 2^{n}} ) for all ( n epsilon ) N. The value of ( a_{1} a_{2} a_{3} dots a_{n} ) is equal to ( A cdot x-y ) в. ( frac{x+y}{b_{n}} ) c. ( frac{x-y}{b_{n}} ) D. ( frac{x y}{b_{n}} ) | 11 |

868 | Find the ( 4^{t h} ) term from the end of the G.P. ( frac{2}{27}, frac{2}{9}, frac{2}{3}, dots dots dots, 162 ) | 11 |

869 | Sum to infinite terms the following series: ( mathbf{1}+mathbf{5} boldsymbol{x}^{2}+mathbf{9} boldsymbol{x}^{4}+mathbf{1} mathbf{3} boldsymbol{x}^{mathbf{6}}+ldots .,|boldsymbol{x}|<mathbf{1} ) | 11 |

870 | Find the mean of the following First eight even natural numbers. | 11 |

871 | ( boldsymbol{S}_{boldsymbol{n}}=boldsymbol{2} boldsymbol{n}^{2}+boldsymbol{3} boldsymbol{n} ; ) then ( boldsymbol{d}=ldots ) A . 13 B. 4 ( c .9 ) D. -2 | 11 |

872 | The given G.P ( 0.15,0.015,0.0015, ldots 20 ) terms can be written as? | 11 |

873 | If ( a, b, c ) be in ( A P, ) and ( a^{2}, b^{2}, c^{2} ) are in ( H ) ( P ., ) then ( mathbf{A} cdot a+b=c ) B. ( b+c=a ) c. ( c+a=b ) ( mathbf{D} cdot a b+b c+c a=0 ) | 11 |

874 | The sum of the infinite series ( 1+ ) ( left(1+frac{1}{5}right)left(frac{1}{2}right)+ ) ( left(1+frac{1}{5}+frac{1}{5^{2}}right)left(frac{1}{2^{2}}right)+dots dots ) A ( cdot frac{20}{9} ) в. ( frac{10}{9} ) ( c cdot frac{5}{9} ) ( D cdot frac{5}{3} ) | 11 |

875 | ( ln operatorname{an} A cdot P ) sum of first ten terms is -150 and the sum of its next ten terms is ( -550, ) Find the ( A . P ) | 11 |

876 | Which term of the A.P., ( 84,80,76, . . ) is ( 0 ? ) A ( cdot 18^{text {th}} ) term B. ( 20^{text {th }} ) term c. ( 22^{text {nd }} ) term D. ( 24^{t h} ) term | 11 |

877 | Find sum of all odd integers between 2 and 100 divisible by 3 | 11 |

878 | The mean of data 34,65,14,74,43 is | 11 |

879 | If ( m ) arithmetic means are inserted between 1 and 31 so that the ratio of the ( 7 t h ) and ( (m-1) t h ) means is ( 5: 9, ) then the value of ( m ) is ( mathbf{A} cdot mathbf{9} ) B. 11 c. 13 D. 14 | 11 |

880 | ( boldsymbol{x}=(-1)^{a^{1}}+(-1)^{a^{2}}+ldots . .(-1)^{a^{1006}} ) ( boldsymbol{y}=(-1)^{a^{1007}}+(-1)^{a^{1008}}+ldots+ ) ( (-1)^{a^{2013}} ) Then which of the following is true? ( mathbf{A} cdot(-1)^{x}=1 ;(-1)^{y}=1 ) B . ( (-1)^{x}=1 ;(-1)^{y}=-1 ) C ( cdot(-1)^{x}=-1 ;(-1)^{y}=1 ) D cdot ( (-1)^{x}=-1 ;(-1)^{y}=-1 ) | 11 |

881 | Compute the geometric mean of 2,4,8 A .4 B. 6 ( c .8 ) D. 2 | 11 |

882 | The sum of the series ( 2left[7^{-1}+3^{1} .7^{-1}+5^{1} .7^{-1}+ldotsright] ) upto 20 twenty terms is A. ( frac{800}{7} ) B. ( frac{580}{7} ) c. ( frac{780}{9} ) D. ( frac{680}{7} ) | 11 |

883 | Find common difference of the following ( 8,15,22,29, dots ) A. 5 B. 6 c. 7 ( D ) | 11 |

884 | If an AP, ratio of the 4 th and 9 th terms is ( 1: 3, ) find the ratio of 12 term and 5 th term? | 11 |

885 | ( 1+0.5+0.25+0.125 ldots . ) is an example of A. finite geometric progression B. infinite geometric series c. finite geometric sequence D. infinite geometric progression | 11 |

886 | Number of real solutions of the equation ( sin a^{x} cos a^{x}=frac{a^{x}+a^{-x}}{4} ) is ( mathbf{A} cdot mathbf{1} ) B . 2 ( c .3 ) D. | 11 |

887 | If the sum of ( 1+frac{1+2}{2}+frac{1+2+3}{3}+ ) ( ldots ) to ( n ) terms is ( S, ) then ( S ) is equal to A ( cdot frac{n(n+3)}{4} ) в. ( frac{n(n+2)}{4} ) c. ( frac{n(n+1)(n+2)}{6} ) D. ( n^{2} ) | 11 |

888 | Product of n positive numbers is unity. The sum of these numbers can not be less than ( mathbf{A} cdot mathbf{1} ) в. ( n ) ( c cdot n^{2} ) D. None of these | 11 |

889 | If nth term of an A.P is ( (2 n+1), ) what is the sum of its first three terms? | 11 |

890 | The number of terms of an ( A . P . ) is even; the sum of the odd terms is ( 24, ) and of the even terms is 30 and the last term exceeds the first by ( 10.5, ) then the number of terms in the series is A . 8 B. 4 ( c .6 ) D. 10 | 11 |

891 | A series is: A. A number of events, objects, or people of a similar or related kind coming one after another. B. Combination of terms following a particular pattern. C. Both A and B D. Non of the above | 11 |

892 | Ram prasad saved ( R s .10 ) in the first week of the year and then increased his weekly saving by ( R s .2 .75 . ) If in the ( n^{t h} ) week, his savings become ( R s .59 .50 ) find ( n ) | 11 |

893 | The value of the largest term common to the sequences ( 1,11,21,31, . . ) upto 100 terms and ( 31,36,41,46, dots ) upto 100 terms, is A . 281 в. 381 c. 471 D. 52 | 11 |

894 | Divide 56 in four parts in A.P. such that the ratio of the product of their extremes (1 st and 4 th) to the product of means ( (2 n d text { and } 3 r d) ) is 5: 6 | 11 |

895 | toppr Q Type your question following steps. Step 1: First of all we find the successive difference (first difference, second difference, third difference ( ldots text { so on }) ) Step 2: If first successive difference is in A.P, then general term can be taken ( operatorname{as} t_{n}=a n^{2}+b n+cleft(text { i.e.consider } t_{n} ) as right. quadratic polynomial in the decreasing power of ( n text { with constants } a, b, c) ) Step 3: If the second, third successive differences are in ( mathbf{A} . mathbf{P}, ) then ( boldsymbol{t}_{boldsymbol{n}}=boldsymbol{a} boldsymbol{n}^{boldsymbol{3}}+ ) ( b n^{2}+c d+d ) for second successive difference in A.P and so on where ( a, b, c, d ) are constants whose order can be changed. Step 4: If the first difference in step 1 is in G.P, then take ( t_{n}=a r^{n-1}+b n+c ) Similarly if second, third difference are in ( G . P, ) then general terms are considered by ( t_{n}=a r^{n-1}+b n^{2}+ ) ( boldsymbol{c n}+boldsymbol{d} ) and ( boldsymbol{t}_{boldsymbol{n}}=boldsymbol{a} boldsymbol{r}^{boldsymbol{n}-mathbf{1}}+boldsymbol{b} boldsymbol{n}^{boldsymbol{3}}+boldsymbol{c n}^{boldsymbol{2}}+ ) ( boldsymbol{d} boldsymbol{n}+boldsymbol{e} ) respectively where ( boldsymbol{a}, boldsymbol{b}, boldsymbol{c}, boldsymbol{d}, boldsymbol{e} ldots ) are constants whose orders can be changed. Now consider the sequences ( P: 9,16,29,54,103, dots ) ( Q: 4,14,30,52,80,114, dots ) ( R: 2,12,36,80,150,252, dots ) ( boldsymbol{S}: mathbf{2}, mathbf{5}, mathbf{1 2}, mathbf{3 1}, mathbf{8 6}, dots ) On the basis of above data (information) answer the following questions:If ( t_{n}=A+B n+C n^{2} ) for sequence ( Q ), then the value of ( A^{2}+ ) ( B^{2}+C^{2} ) equals A. 10 B. 8 ( c . ) D. None of these | 11 |

896 | A sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence is known as: A. geometric series B. arithmetic progression c. harmonic sequence D. geometric sequence | 11 |

897 | Three numbers ( x, y ) and ( z ) are in arithmetic progressions. If ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}= ) -3 and ( x y z=8, ) then ( x^{2}+y^{2}+z^{2} ) is equal to ( A cdot 0 ) B. 10 ( c cdot 21 ) D. 20 | 11 |

898 | The sum of the three numbers which are in arithmetic progression is ( 12 . ) If the third number is three times the first number, than the product of the numbers is A . 32 B. 40 c. 24 D. 48 | 11 |

899 | The number of terms is given by in the expression ( t_{n}=a+ ) ( (n-1) d ) A ( cdot t_{n} ) B. ( n ) ( c cdot a ) D. ( d ) | 11 |

900 | Find the sum of n terms of GP: ( sqrt{7}, sqrt{21}, 3 sqrt{7}, dots n ) terms. | 11 |

901 | Find the ( 31^{s t} ) term of an AP whose ( 11^{t h} ) term is 38 and the ( 16^{t h} ) term is 73 | 11 |

902 | The ratio between the sum of ( n ) terms of two A.P.’s is ( 7 n+1: 4 n+27 . ) Find the ratio between their nth terms. | 11 |

903 | In an A.P. the sum of terms equidistant from the beginning and end is equal to? | 11 |

904 | (2001 – 5 Marks) Let a, b be positive real numbers. If a, A1, A2, b are in arithmetic progression, a, G, G , b are in geometric progression and a, H, H2, b are in harmonic progression, GG2 4 + A2 (2a +b)(a +2b) show that H H H + H 9 ab 200 ml | 11 |

905 | If ( a, b ) and ( c ) are in geometric progression, then ( a^{2}, b^{2} ) and ( c^{2} ) are in progression. ( A cdot A P ) в. GР c. нр D. AGP | 11 |

906 | State True or False. ( boldsymbol{n}^{n}>1 times mathbf{3} times mathbf{5} times mathbf{7} times ldots ldots times(mathbf{2 n}-mathbf{1}) ) A. True B. False | 11 |

907 | If ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y}, boldsymbol{x}-boldsymbol{y})=boldsymbol{x} boldsymbol{y}, ) then the arithmetic mean of ( boldsymbol{f}(boldsymbol{x}, boldsymbol{y}) ) and ( boldsymbol{f}(boldsymbol{y}, boldsymbol{x}) ) is | 11 |

908 | The first term of a G.P. is ( 1 . ) The sum of the third and fifth term is ( 90 . ) Find the common ratio of the G.P. | 11 |

909 | The sum of the series ( 10-5+2.5 ) ( 1.25 ldots . . ) is called A . finite geometric sequence B. finite arithmetic sequence C. infinite geometric sequence D. infinite harmonic sequence | 11 |

910 | Three positive real numbers ( l, boldsymbol{m}, boldsymbol{n} ) are in A.P. with product taken all at a time is ( 4, ) then the minimum value of ( m ) is ( mathbf{A} cdot 4^{1 / 3} ) B. 3 ( c cdot 2 ) D. ( 1 / 2 ) | 11 |

911 | The arithmetic mean of ( 1+sqrt{2} ) and ( 7+ ) ( mathbf{5} sqrt{mathbf{2}} ) is ( sqrt{boldsymbol{a}}+sqrt{boldsymbol{b}} . ) Then ( boldsymbol{a}-boldsymbol{b}= ) A . – B. ( c cdot 2 ) ( D cdot-2 ) | 11 |

912 | How many terms are there in the G.P ( mathbf{3}, mathbf{6}, mathbf{1 2}, mathbf{2 4}, dots dots dots, mathbf{3 8 4} ? ) ( mathbf{A} cdot mathbf{8} ) B. 9 c. 10 D. 11 E. 7 | 11 |

913 | Determine the relations in ( x, y ) and ( z ) if ( 1, log _{y} x, log _{z} y,-15 log _{x} z ) are in A.P. | 11 |

914 | Sum the following series to n terms: ( mathbf{3}+mathbf{5}+mathbf{9}+mathbf{1 5}+mathbf{2 3}+dots ) A ( cdot frac{n}{3}left(n^{2}-8right) ) В ( cdot frac{n}{3}left(n^{2}+8right) ) c. ( frac{n}{2}left(n^{3}+8right) ) D. None of these | 11 |

915 | The sum of four consecutive terms which are in an arithmetic progression is 32 and the ratio of the product of the first and the last term to the product of two middle terms is 7: 15 . Find the number. | 11 |

916 | The nth term of the A.P. is ( 2 n-5 ), then the series is : A. ( -3,-1,1, ldots ) . в. ( 3,1,-1, ldots ) c. ( 2,5,8, dots dots ) D. ( 1,7,13, ldots ldots ) | 11 |

917 | Find the sum of the following arithmetic progressions: ( a+b, a-b, a-3 b, ldots . . ) to 22 terms. | 11 |

918 | If the heights of 5 persons are ( 144 mathrm{cm} ) ( 152 mathrm{cm}, 151 mathrm{cm}, 158 mathrm{cm}, ) and ( 155 mathrm{cm} ) respectively. Find the mean height. | 11 |

919 | If ( 9^{t h} ) term of an ( A . P . ) is zero, Prove that its ( 29^{t h} ) term is double the ( 19^{t h} ) term. | 11 |

920 | since the beginning of ( 1990, ) the number of squirrels in a certain wooded area has tripled during every 3-year period of time. If there were 5,400 squirrels in the wooded area at the beginning of ( 1999, ) how many squirrels were in the wooded area at the beginning of ( 1990 ? ) A . 50 B. 100 ( c cdot 200 ) D. 300 | 11 |

921 | The AP whose first term is 10 and common difference is 3 is B. ( 5,7,9,11, ldots ) c. ( 8,12,16,20, dots ) D. All the above | 11 |

922 | Find the sum of all multiples of 5 lying between 101 and 999 | 11 |

923 | If ( boldsymbol{x}=log _{5} boldsymbol{3}+log _{7} mathbf{5}+log _{9} boldsymbol{7} ) then A ( cdot x geq frac{3}{2} ) в. ( x geq frac{1}{sqrt[3]{2}} ) c. ( x geq frac{3}{sqrt[3]{2}} ) D. none of these | 11 |

924 | The geometric mean of the numbers ( mathbf{7}, mathbf{7}^{2}, mathbf{7}^{mathbf{3}}, dots, mathbf{7}^{n} mathbf{i} mathbf{s} ) ( A cdot 7^{7 / 4} ) B ( cdot 7^{4 / 7} ) ( c cdot 7^{frac{n-1}{2}-frac{1}{2}-frac{2}{2}-frac{2}{2}} ) ( D cdot 7^{frac{n-1}{2}} ) | 11 |

925 | Let ( a_{1}, a_{2}, a_{3}, dots ) be terms of an A.P. If ( frac{boldsymbol{a}_{1}+boldsymbol{a}_{2}+ldots . .+boldsymbol{a}_{p}}{boldsymbol{a}_{1}+boldsymbol{a}_{2}+ldots . .+boldsymbol{a}_{boldsymbol{q}}}=frac{boldsymbol{p}^{2}}{boldsymbol{q}^{2}}, boldsymbol{p} neq boldsymbol{q}, ) then ( frac{u_{6}}{2} ) equals | 11 |

926 | The AM of two given positive number is 3. If the larger number is increased by 1 the GM of a numbers becomes equal to AM of the given numbers. Then the HM of the given numbers is A ( cdot frac{3}{2} ) B. ( frac{2}{3} ) ( c cdot frac{1}{2} ) D. None of these | 11 |

927 | Given ( l=28, S=144 ) and there are total 9 terms. Find ( a ) | 11 |

928 | If ( 3+5+7+9+ldots ) upto ( n ) terms ( =288 ) then ( n= ) A . 12 B . 15 c. 16 D. 17 | 11 |

929 | Find the sum of all odd natural numbers less than 50 . | 11 |

930 | Which term of the G.P. ( 2,8,32, ldots ) is ( 131072 ? ) | 11 |

931 | The sum of the series ( 1+frac{1}{4 times 2 !}+ ) ( frac{1}{16 times 4 !}+frac{1}{64 times 6 !}+ldots infty ) is? A ( cdot frac{e+1}{sqrt{e}} ) в. ( frac{e-1}{sqrt{e}} ) c. ( frac{e+1}{2 sqrt{e}} ) D. ( frac{e-1}{2 sqrt{e}} ) | 11 |

932 | Find the sum of 32 terms of an A.P. whose third terms is 1 and 6 th term is -11 | 11 |

933 | Sum of the first ( p, q ) and ( r ) terms of an A.P. are ( a, b ) and ( c, ) respectively. Prove that ( frac{boldsymbol{a}(boldsymbol{q}-boldsymbol{r})}{boldsymbol{p}}+frac{boldsymbol{b}(boldsymbol{r}-boldsymbol{p})}{boldsymbol{q}}+ ) ( frac{c(p-q)}{r}=0 ) | 11 |

934 | Assertion ( a, b, c ) are three unequal positive numbers. STATEMENT-1: The product of their sum and the sum of their reciprocals exceeds 9 Reason STATEMENT-2: AM of ( n ) positive numbers exceeds their HM. A. Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement- B. Statement-1 is True, Statement-2 is True; Statement-2 is Not a correct explanation for Statement- c. Statement-1 is True, Statement-2 is False D. Statement-1 is False, Statement-2 is True | 11 |

935 | If ( S_{r} ) denotes the sum of ( r ) terms of an ( A P ) and ( frac{S_{a}}{a^{2}}=frac{S_{b}}{b^{2}}=c ) then ( S_{c} ) is A ( . c^{3} ) в. ( c / a b ) ( c cdot a b c ) D. ( a+b+c ) | 11 |

936 | If ( a_{n} ) is a sequence such that ( a_{1}=1 ) ( boldsymbol{a}_{2}=mathbf{1}+boldsymbol{a}_{1}, boldsymbol{a}_{3}=mathbf{1}+boldsymbol{a}_{1} boldsymbol{a}_{2}, boldsymbol{a}_{n+1}= ) ( 1+a_{1} a_{2} a_{3} dots dots a_{n} ) then find the value of ( frac{1}{a_{1}}+frac{1}{a_{2}}+frac{1}{a_{3}}+frac{1}{a_{4}} dots dots ) | 11 |

937 | If ( a_{1}, a_{2}, a_{3}, dots dots dots ) be in ( A . P . ) such that ( boldsymbol{a}_{1}+boldsymbol{a}_{5}+boldsymbol{a}_{10}+boldsymbol{a}_{15}+boldsymbol{a}_{20}+boldsymbol{a}_{24}=mathbf{2 2 5} ) then the sum of first 24 terms of the A.P is | 11 |

938 | Find ( a_{20} ) given that ( a_{3}=frac{1}{2} ) and ( a_{5}=8 ) (Given : The terms are in A.P.) | 11 |

939 | Find the sum of series up to ( n ) terms: ( frac{1^{3}}{1}+frac{1^{3}+2^{3}}{1+3}+frac{1^{3}+2^{3}+3^{3}}{1+3+5}+dots ) | 11 |

940 | Which term of an A.P ( 7,3,-1, ldots ) is ( -89 ? ) | 11 |

941 | If 100 times the 100 th term of an A.P. with non-zero common difference equals the 50 times its 50 th term. Then 150th term of thus A.P. is A . -150 B. 150 times the soth term ( c cdot ) 150 D. | 11 |

942 | The harmonic mean and geometric mean of two positive numbers are in the ratio ( 4: 5, ) then two numbers are in the ratio A . 4: 1 B. 3: 1 ( c cdot 2: 1 ) ( D .5: ) | 11 |

943 | If ( S_{1}, S_{2}, S_{3} ) be respectively the sums of ( n, 2 n, 3 n ) terms of a ( G . P ., ) then prove that If the series ( a_{1}, a_{2}, a_{3}, a_{n} ) be a ( G . P . ) of | 11 |

944 | ( a_{n}=frac{2 n-3}{6} ) | 11 |

945 | The number of positive integral solutions of ( a b c d=210 ) is A . 16 B. 64 ( c cdot 256 ) D. 1028 | 11 |

946 | How will you identify the sequence is an infinite geometric progression? A. An geometric sequence containing finite number of terms B. An geometric sequence containing infinite number of terms C. An arithmetic sequence containing infinite number of terms D. An arithmetic sequence containing finite number of terms | 11 |

947 | If ( m ) and ( n ) are positive real numbers and logm, ( log left(frac{m^{2}}{n}right), log left(frac{m^{2}}{n^{2}}right) ) are in A.P then its general term is- ( ^{mathbf{A}} cdot log left(frac{m^{r}}{n^{r-1}}right) ) ( ^{mathbf{B}} cdot log left(frac{m^{r+1}}{n^{r}}right) ) ( ^{mathbf{c}} cdot log left(frac{m}{n}right)^{n} ) D. ( log left(frac{m^{r-1}}{n^{r+1}}right) ) | 11 |

948 | Number of identical terms in the sequence ( 2,5,8,11 ldots ) upto 100 terms are A . 17 B. 33 c. 50 D. 147 | 11 |

949 | The first two terms in a GP as 3 and 6 what is the ( 10^{t h} ) term? A . 1536 B . 2536 ( c .3536 ) D. 4536 | 11 |

950 | The first and second terms of both an A.P. and a G.P. are same ( x ) and ( y ) respectively where both ( x ) and ( y ) are ( +i v e ) and ( x ) is greater than ( y . ) If ( S ) be the sum of infinite ( G . P ., ) then prove that sum of first ( n ) terms of ( A . P . ) is given by ( boldsymbol{n x}-frac{boldsymbol{n}(boldsymbol{n}-mathbf{1}) boldsymbol{x}^{2}}{boldsymbol{2} boldsymbol{S}} ) | 11 |

951 | Check whether given series is ( A P s ? ) If they form an ( A P ), find the common difference ( d ) and write three more terms. ( -1.2,-3.2,-5.2,-7.2, dots . ) | 11 |

952 | The value of ( frac{1}{(2 n-1) ! 0 !}+ ) ( frac{1}{(2 n-3) ! 2 !}+frac{1}{(2 n-5) ! 4 !}+dots+ ) ( frac{1}{1 !(2 n-2) !} ) equal to A ( cdot 2^{2 n-1} ) B. ( 2^{2 n-2} ) ( mathrm{c} cdot 2^{2 n-3} ) D. ( frac{2^{2 n-2}}{(2 n-1) !} ) | 11 |

953 | The 17 th term of the series ( 3+7+11+15 ) + ………… is: ( A cdot 63 ) B. 65 ( c cdot 67 ) D. 69 | 11 |

954 | There is an auditorium with 35 rows of seats. There are 20 seats in the first row, 22 seats in the second row, 24 seats in the third row and so on. Find the number of seats in the twenty-third row. | 11 |

955 | Sum the series: ( 1-frac{1}{3}+frac{1}{3^{2}}-frac{1}{3^{3}}+frac{1}{3^{4}} dots dots infty ) ( A cdot frac{3}{4} ) B. ( frac{4}{3} ) ( c cdot frac{2}{3} ) D. 3 | 11 |

956 | ( operatorname{Let} boldsymbol{S}= ) ( left{frac{a^{2}+b^{2}+c^{2}}{a b+b c+c a}: a, b, c epsilon R, a b+b c+c aright. ) where ( R ) is the set of real numbers. Then S equals. A ( cdot(-infty,-1] cup[1, infty) ) В . ( (-infty, 0) cup(0, infty) ) c. ( (-infty,-1] cup[2, infty) ) D ( cdot(-infty,-2] cup[1, infty) ) | 11 |

957 | If ( f(x) ) is a differentiable function in the interval ( (0, infty) ) such that ( f(1)=1 ) and ( lim _{t rightarrow x} frac{t^{2} f(x)-x^{2} f(t)}{t-x}=1 ) A ( cdot frac{25}{9} ) в. ( frac{23}{18} ) c. ( frac{13}{6} ) D. ( frac{31}{18} ) | 11 |

958 | The least length of the thread required to construct a rectangle of area 256 ( c m^{2} ) is A . 32 B. 64 ( c cdot 40 ) D. 58 | 11 |

959 | Find the value of ( lim _{n rightarrow infty} frac{1}{n} sum_{r=1}^{n} sin ^{2} frac{r pi}{n} ) | 11 |

960 | Arithmetic mean of 2 and 8 is ( mathbf{A} cdot mathbf{5} ) B. 10 c. 16 D. 3.2 | 11 |

961 | If the nth term of the A.P is ( 6 n-1 ) find the ( boldsymbol{S}_{boldsymbol{n}} ) | 11 |

962 | What is the value of ( frac{m^{2}+m n}{m^{2}+n^{2}}+ ) ( frac{m^{2}-m n}{m^{2}+n^{2}}+frac{n^{2}+m}{m^{2}+n^{2}}+frac{n^{2}-m n}{m^{2}+n^{2}} ) ( A ) B. 0 ( c cdot 2 ) ( D ) | 11 |

963 | 60. The next term of the sequence 1, 9, 28, 65, 126, … is (1) 199 (2) 205 (3) 216 (4) 217 | 11 |

964 | Find the sum: ( sqrt{2}+sqrt{8}+sqrt{18}+ ) ( sqrt{32} ldots . . ) up to ( n ) terms | 11 |

965 | If ( a_{1}, a_{2}, a_{3} dots ) are in A.P.with common difference ‘d’,then ( tan left{tan ^{-1}left(frac{d}{1+a_{1} a_{2}}right)+tan ^{-1}left(frac{d}{1+a_{2} a_{3}}right)+right. ) is equal to A ( cdot frac{(n-1) d}{a_{1}+a_{n}} ) B ( cdot frac{(n-1) d}{1+a a_{n}} ) c. ( frac{(n-1) d}{1+a a a_{n}} ) D. ( frac{a_{n}-a_{1}}{a_{n}+a_{1}} ) | 11 |

966 | An Arithmetic progression consists of 20 terms of which 4 th term is 16 and the last term is ( 208 . ) Find the 15 th term A . 146 B. 147 ( c cdot 148 ) D. 149 | 11 |

967 | Find the first term and common difference in the given sequence: ( frac{3}{2}, frac{1}{2},-frac{1}{2},-frac{3}{2}, dots ) | 11 |

968 | In any ( Delta A B C, sumleft(frac{sin ^{2} A+sin A+1}{sin A}right) ) is always greater than A . 9 B. 3 c. 27 D. | 11 |

969 | How many terms of the AP: ( 15,13,11, ) are needed to make the sum 55? Explain the reason for double answer. | 11 |

970 | Find the geometric mean between 16 and 81 A . 4 B. 9 ( c .36 ) D. 18 | 11 |

971 | Find ( n ) if the given value of ( x ) is the ( n ) th term of the given A.P. ( 25,50,75,100, dots dots ; x=1000 ) | 11 |

972 | Write the first three terms of the AP when a and d are as given below: ( boldsymbol{a}=mathbf{5}, boldsymbol{d}=mathbf{3}, ) then first three terms are ( mathbf{5}, mathbf{8}, mathbf{1 1} ) If true then enter 1 and if false then enter 0 | 11 |

973 | Let ( S_{n} ) denote the sum of first ( n ) terms of an AP and ( 3 S_{n}=S_{2 n} . ) What is ( S_{3 n}: S_{2 n} ) equal to? A .2: 1 B. 3: 1 c. 4: 1 D. 5: 1 | 11 |

974 | A sequence is defined by ( a_{n}=n^{3}- ) ( 6 n^{2}+11 n-6 . ) Show that the first three terms of the sequence are zero and al other terms are positive. | 11 |

975 | The sum of the first ( n ) terms of a sequence is ( frac{7^{n}-6^{n}}{6^{n}}, ) Find its ( n^{t h} ) term Determine whether the sequence is A.P. or G.P | 11 |

976 | Let a sequence be defined by ( a_{1}= ) ( mathbf{1}, boldsymbol{a}_{mathbf{2}}=mathbf{1} ) and ( , boldsymbol{a}_{boldsymbol{n}}=boldsymbol{a}_{boldsymbol{n}-mathbf{1}}+boldsymbol{a}_{boldsymbol{n}-mathbf{2}} ) for al ( boldsymbol{n}>2 ) find ( frac{boldsymbol{a}_{boldsymbol{n}+1}}{boldsymbol{a}_{boldsymbol{n}}} ) for ( boldsymbol{n}=mathbf{1}, boldsymbol{2}, boldsymbol{3}, boldsymbol{4} ) | 11 |

977 | What is the function for the arithmetic sequence ( 3,4,5,6,7 dots ? ) ( mathbf{A} cdot n+2 ) в. ( n-1 ) c. ( 2 n+1 ) D. ( 2 n-1 ) | 11 |

978 | Find the value of a if ( 34,45,56, a ) if they are in AP A . 67 B. 66 ( c cdot 65 ) D. 69 | 11 |

979 | If ( b_{i}=1-a_{i}, n a=sum_{i=1}^{n} a_{i}, n b= ) ( sum_{i=1}^{n} b_{i} quad, ) then ( sum_{i=1}^{n} a_{i} b_{i}+ ) ( sum_{i=1}^{n}left(boldsymbol{a}_{i}-boldsymbol{a}right)^{2}= ) ( mathbf{A} cdot a b ) B. ( -n a b ) c. ( n a b ) D. ( (n+1) a b ) | 11 |

980 | Find the average of first twelve natural numbers. | 11 |

981 | which term of the A.P. ( mathbf{1 0 0}, mathbf{9 7}, mathbf{9 4}, mathbf{9 1}, ldots ) is its first negative term? | 11 |

982 | When each term of a sequence is connected using ( a+ ) or ( a-operatorname{sign}, ) then it is referred to as the of numbers A. Series B. Progression c. Arithmetic Progression D. Geometric Prpgression | 11 |

983 | 25. Given that the side length of a rhombus is the geometric mean of the lengths of its diagonals. The degree measure of the acute angle of the rhombus is a. 15° b. 30° c. 45° d. 60° | 11 |

984 | Find the sum of all natural numbers lying between 100 and ( 1000, ) which are multiples of 5 | 11 |

985 | The sum of first ( p ) -terms of a sequence is ( p(p+1)(p+2) . ) The ( 10^{t h} ) term of the sequence is A . 396 в. 600 ( c .330 ) D. 114 | 11 |

986 | Find the sum of ( n ) terms.Also find the sum to infinite terms: ( 1+frac{1}{1+2}+frac{1}{1+2+3}+dots ) | 11 |

987 | ( a_{1}=-2, r=-1 . ) Find the ( 5^{t h} ) term in GP. ( mathbf{A} cdot mathbf{1} ) B. – ( c cdot 2 ) D. – 2 | 11 |

988 | Write the common difference of the A.P. ( 7,5,3,1,-1,-3, dots ) | 11 |

989 | If ( sin theta ) is ( mathrm{G} . mathrm{M} ) of ( sin phi ) and ( cos phi, ) then prove that ( cos 2 theta=2 cos ^{2}left(frac{pi}{4}+phiright) ) | 11 |

990 | If four number in A.P are such that their sum is 50 and the greatest number is 4 times the least then the number are ( A cdot 5,10,15,20 ) B. 4, 10, 16, 22 c. 3,7,11,15 D. None of these | 11 |

991 | Is the given sequence an AP? If it forms an ( A P, ) find the common difference ( d ) and write the next three terms. ( mathbf{0},-mathbf{4},-mathbf{8},-mathbf{1 2} dots ) | 11 |

992 | Solve: ( mathbf{5}+mathbf{5}^{2}+mathbf{5}^{3}+ldots ldots+mathbf{5}^{n} ) | 11 |

993 | Sum of infinity the following series: ( mathbf{1}+mathbf{4} boldsymbol{x}^{mathbf{2}}+mathbf{7} boldsymbol{x}^{mathbf{4}}+mathbf{1 0} boldsymbol{x}^{mathbf{6}}+ldots,|boldsymbol{x}|<mathbf{1} ) | 11 |

994 | Prove ( n^{n}>1.3 .5, dots . . .(2 n-1) ) | 11 |

995 | Find the sum of ( n ) terms of the series ( 3+8+22+72+266+1036+dots ) A ( cdot frac{3 n}{4}+n(n+1)+frac{1}{12}(4 n-1) ) B . ( frac{3 n}{4}+n(n+1)+frac{1}{12}left(4^{n-1}right) ) c. ( frac{3 n}{4}+n(n+1)+frac{1}{12}(4 n-2) ) D. ( frac{3 n}{4}+n(n+1)+frac{1}{12}left(4^{n-2}right) ) | 11 |

996 | The value of ( p ) if ( 3, p, 12 ) are in GP A. 6 B. 4 ( c .9 ) D. None | 11 |

997 | Le S, S,, …… be squares such that for each n 21, the length of a side of S equals the length of a diagonal of Sn+ 1 If the length of a side of s, is 10 cm, then for which of the following values of n is the area of S less than 1 sq. cm? (1999 – 3 Marks) (a) 7 (6) 8 (c) 9 (d) 10 . 5 let N be the numbe | 11 |

998 | If ( A ) be the ( A . M . ) and ( H ) the H.M. between two numbers ( a ) and ( b ), then show ( frac{boldsymbol{a}-boldsymbol{A}}{boldsymbol{a}-boldsymbol{H}} times frac{boldsymbol{b}-boldsymbol{A}}{boldsymbol{b}-boldsymbol{H}}=frac{boldsymbol{A}}{boldsymbol{H}} ) | 11 |

999 | In a certain ( A . P ., 5 ) times the 5 th term is equal to 8 times the 8 th term, then find its 13 th term. | 11 |

1000 | Sum of certain number of terms of the series ( frac{2}{9},-frac{1}{3}, frac{1}{2}, dots ) is ( frac{55}{72} . ) Find the number. | 11 |

1001 | The greatest value of ( a b^{3} c ) is ( A cdot 3 ) B. 9 c. 27 D. 81 | 11 |

1002 | The harmonic mean of two numbers is 4. Their arithmetic mean is A and geometric mean is G. If G satisfies ( 2 A+G^{2}=27, ) the numbers are A .1,13 в. 9,12 c. 3,6 D. 4,8 | 11 |

1003 | If sum of ( n ) terms of A.P. is ( 476, ) last term ( =20, n=17, ) then the first term is A .32 B. 34 c. 36 D. 38 8 | 11 |

1004 | Sum ( 1.3,-3.1,-7.5, dots ) to 10 terms | 11 |

1005 | Which term of the arithmetic progression ( 5,15,25, ldots ) will be 130 than its ( 31^{s t} ) term? | 11 |

1006 | If ( a, b, c ) are sides of the ( triangle A B C ) such that ( left(1+frac{b-c}{a}right)^{a} cdotleft(1+frac{c-a}{b}right)^{b} ) ( left(1+frac{a-b}{c}right)^{c} geq 1, ) then triangle ( triangle A B C ) must be A. right angled B. isosceles c. obtuse D. equilateral | 11 |

1007 | Find the sum of ( n ) terms. Also find the sum of infiite terms ( 1+frac{1}{1+2}+frac{1}{1+2+3}+dots ) | 11 |

1008 | ( a, b, c, d, e ) are in ( A . P . ) Prove the following results ( boldsymbol{a}-boldsymbol{2} boldsymbol{b}+boldsymbol{c}=mathbf{0} ) | 11 |

1009 | Find the sum of the series ( frac{3 cdot 5}{5 cdot 10}+frac{3 cdot 5 cdot 7}{5 cdot 10 cdot 15}+frac{3 cdot 5 cdot 7 cdot 9}{5 cdot 10 cdot 15 cdot 20}+ ) ( dots infty ) | 11 |

1010 | Let angle ( A, B ) and ( C ) of a triangle ( A B C ) be in arithmetic progression. If ( frac{b}{c}=sqrt{frac{3}{2}} ) then the value of cosec ( 2 A ) is equal to (Symbols used have usual meaning in a triangle ( A B C ) ) | 11 |

1011 | Find a wrong number in the series: 9,19,37,75,149,297 A . 75 B. 37 c. 19 D. 149 E . 299 | 11 |

1012 | 12. Let the positive numbers a, b, c, d be in A.P. Then abc, abd, acd, bcd are (20015) (a) NOT in A.P./GP./H.P. (b) in A.P. (c) in GP. (d) in H.P. | 11 |

1013 | Find 10 th term of the ( A . P .1,4,7,10, dots ) | 11 |

1014 | Find the common ratio and the general term of the following geometric sequences. ( frac{2}{5}, frac{6}{25}, frac{18}{125}, dots ) | 11 |

1015 | 1. If1, log, (31-x+2), log; (4.34 – 1) are in A.P. then x equals [2002] (a) log, 4 (b) 1-logz 4 (d) log 3 (c) 1- logạº 10843 4 th 1 | 11 |

1016 | The arithmetic mean (average) of a set of 50 numbers is ( 38 . ) If two numbers, namely, 45 and ( 55, ) are discarded, the mean of the remaining set of numbers is : A . 36.5 B. 37 c. 37.2 D. 37.5 E . 37.52 | 11 |

1017 | ( operatorname{any} Delta A B C, Pileft(frac{sin ^{2} A+sin A+1}{sin A}right) ) is always greater than This question has multiple correct options ( A cdot 9 ) B. 3 c. 27 D. 81 | 11 |

1018 | Four numbers are inserted between the numbers 4 and 39 such that an ( A P ) results. Find”the biggest of these four numbers. A . 33 B. 31 ( c cdot 32 ) D. 30 | 11 |

1019 | Match the List I with the List II. | 11 |

1020 | 24. If a, b, c, d are positive real numbers such that a+b+c+d=2, then M=(a+b) (c+d) satisfies the relation (a) 0<Ms1 (b) 15M2 (2000) (c) 2SM <3 (d) 35 M 34 | 11 |

1021 | (d) 232 36. If a, b and c be three distinct real num a+b+c=xb, then x cannot be: (a) – 2 (b) (c) 4 (d) distinct real numbers in G.P. and nen x cannot be: TJEEM 2019-9 Jan (M) -3 | 11 |

1022 | The ( 3^{r d} ) term of an A.P. is -40 and ( 13^{t h} ) term is zero, then ( d ) is equal to : A . -4 B. 4 c. 0 D. – | 11 |

1023 | Find the sum of first 25 terms of an AP whose nth term is ( 1-4 n ) | 11 |

1024 | Find four consecutive terms of an A.P. whose sum is 88 and sum of first and third term is 40 | 11 |

1025 | The sum of first 4 term of GP with ( a= ) ( mathbf{2}, boldsymbol{r}=mathbf{3} ) is A . 80 B . 26 ( c cdot 127 ) D. 8 | 11 |

1026 | Four terms are in ( A . P . ) If sum of numbers is 50 and largest number is four times the smaller one, then find the terms. | 11 |

1027 | Find the common difference of an AP. whose first term is 100 and the sum of whose first six terms is five times the sum of the next six terms. A . 10 B. -10 ( c .5 ) D. – 5 | 11 |

1028 | The value of the sum ( sum_{n=1}^{13}left(i^{n}+i^{n+1}right) ) where ( boldsymbol{i}=sqrt{-mathbf{1}} ) is: ( mathbf{A} cdot i ) B. ( -i ) ( c cdot 0 ) D. ( i-1 ) | 11 |

1029 | If non zero numbers ( a, b, c ) are in A.P. then ( a+frac{1}{b c}, b+frac{1}{c a}, c+frac{1}{a b} ) are in : A. G.P B. Н.Р. ( c . ) A.P D. None of these | 11 |

1030 | The point ( P(a, b) ) is such that ( b- ) ( 25 a=4 ) and the arithmetic mean of ( a ) an ( b ) is ( 28 . Q(x, y) ) is the point such that ( x ) and ( y ) are two geometric means between ( a ) and ( b ) if ( O ) is the origin then ( O P^{2}+O Q^{2} ) is equal to | 11 |

1031 | If ( frac{1}{1^{2}}+frac{1}{2^{2}}+frac{1}{3^{2}}+ldots . . ) upto ( infty=frac{pi^{2}}{6} ) then ( frac{1}{1^{2}}+frac{1}{3^{2}}+frac{1}{5^{2}}+ldots= ) A ( cdot frac{pi^{2}}{12} ) в. ( frac{pi^{2}}{24} ) c. ( frac{pi^{2}}{8} ) D. ( frac{pi^{2}}{4} ) | 11 |

1032 | If positive numbers ( x, y, z ) are in ( A . P ) then the minimum value of ( frac{boldsymbol{x}+boldsymbol{y}}{mathbf{2} boldsymbol{y}-boldsymbol{x}}+ ) ( frac{y+z}{2 y-z} ) is equal to | 11 |

1033 | Identify the series ( frac{1}{3}, frac{5}{3}, frac{9}{3}, frac{13}{3}, ) | 11 |

1034 | In a triangle ( A B C ) ( operatorname{acos}^{2}left(frac{C}{2}right)+c cos ^{2}left(frac{A}{2}right)=frac{3 b}{2}, ) then the ( operatorname{sides} a, b, c ) A. Satisfy ( a+b=c ) B. are in ( A . P ). c. are in ( G . P ) D. are in ( H . P ) | 11 |

1035 | Find the common ratio in the following G.P. ( sqrt{mathbf{3}}, mathbf{3}, mathbf{3} sqrt{mathbf{3}} ) | 11 |

1036 | Identify whether the following sequence is a geometric sequence or not. 2,6,18,54 | 11 |

1037 | For what values of ( k ) will be ( k+9,2 k- ) 1 and ( 2 k+7 ) are the consecutive terms of AP. | 11 |

1038 | Assertion If ( a, b, c ) each greater than zero are in A.P. then minimum value of ( b=4 ) if ( a b c=64 ) Reason ( boldsymbol{A} cdot boldsymbol{M} geq boldsymbol{G} cdot boldsymbol{M} ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion C. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect | 11 |

1039 | An an arithmetic progression of 50 terms, the sum of first ten terms is 210 and the sum of last fifteen terms is 2565. Find the arithmetic progression. | 11 |

1040 | Express ( 25 sin h x-24 sin h x ) in the form ( R cos h 2(x-alpha) ) giving the values of ( boldsymbol{R} ) and ( tan boldsymbol{h} boldsymbol{alpha} ) Hence write down the minimum value of ( 25 cos h x-24 sin h x ) and find the value of ( x ) at which this occurs, and give your answer in terms of a natural logarithms. | 11 |

1041 | ( 3^{r d} ) term of a G.P. is 27 and its ( 6^{t h} ) term is ( 729 . ) Find the product of its first and ( mathbf{7}^{t h} ) terms. | 11 |

1042 | Find the least number of terms of an A.P., ( 64+49+34+dots dots ) to be added so that the sum is less than 36 | 11 |

1043 | If each entry of a data is decreased by 8 what is the change in the arithmetic mean? A. Remains the same B. Decreases by 8 C. Increases by 8 D. Can’t be determined | 11 |

1044 | If there are ‘ ( n ) ‘ arithmetic means between ( a ) and ( b ), then the common difference ( (boldsymbol{d})= ) A ( cdot frac{a-b}{n+1} ) в. ( frac{b-a}{n+1} ) c. ( frac{a+b}{n-1} ) D. ( frac{b-a}{n-1} ) | 11 |

1045 | Is the sequence ( sqrt{mathbf{3}}, sqrt{mathbf{6}}, sqrt{mathbf{9}}, sqrt{mathbf{1 2}}, ldots ldots ) from an Arithmetic Progression?Give reason. | 11 |

1046 | If the sum of first ( p ) terms of an ( A . P . ) is equal to the sum of the first ( q ) terms, then find the sum of the first ( (boldsymbol{p}+boldsymbol{q}) ) terms. | 11 |

1047 | The infinite sum ( 1+ ) ( frac{4}{7}+frac{9}{7^{2}}+frac{16}{7^{3}}+frac{25}{7^{4}}+dots dots ) equals A ( cdot frac{27}{14} ) в. ( frac{21}{13} ) c. ( frac{49}{27} ) D. ( frac{256}{147} ) | 11 |

1048 | The minimum value of ( p x+q y ) when ( boldsymbol{x} boldsymbol{y}=boldsymbol{r}^{2} ) is A ( .2 r sqrt{p q} ) B. ( r sqrt{p q} ) c. ( r sqrt{q / p} ) D. None of the above | 11 |

1049 | Find the common ratio of GP whose first term is ( 3, ) the last is 3072 and the sum of the series is 4095 A .2 B. 3 ( c cdot 4 ) D. 6 | 11 |

1050 | The ( 4^{t h} ) term of an AP is 14 and its ( 12^{t h} ) term is ( 70 . ) What is its first term? A . -10 B. -7 ( c cdot 7 ) D. 10 | 11 |

1051 | Identify the formula for the ( n^{t h} ) term of the sequence ( 54,18,6 dots ) ( ^{mathrm{A}} cdot_{54}left(frac{1}{3}right)^{n-1} ) ( ^{text {В }} ). ( _{6}left(frac{1}{3}right)^{n-1} ) c. ( _{3}left(frac{1}{3}right)^{n} ) D. ( _{54}left(frac{1}{3}right)^{n} ) | 11 |

1052 | The sum of the series ( 1-frac{3}{2}+frac{5}{4}-frac{7}{8}+ ) ( ldots infty ) is ( ^{A} cdot frac{2}{9} ) B. ( frac{-4}{9} ) c. ( frac{4}{9} ) D. ( frac{-2}{9} ) | 11 |

1053 | Check if given series is ( A P ) or not? If they form an ( A P ), find the common difference ( d ) and write three more terms. ( 0.2,0.22,0.222,0.2222, ldots . ) | 11 |

1054 | f ( a_{1}, a_{2}, a_{3}, a_{4} ) are the terms in AP. if ( a_{1}=5 ; d=9, a_{4} ) is equal to A . 32 B. 27 c. 25 D. 22 | 11 |

1055 | The sum of 7 terms of the series ( 1^{2}- ) ( mathbf{2}^{2}+mathbf{3}^{2}-mathbf{4}^{2}+mathbf{5}^{2}-mathbf{6}^{2}+dots ) is A . -21 B. 15 c. 28 D. -35 | 11 |

1056 | Evaluate: ( mathbf{2}+mathbf{2}^{mathbf{2}}+mathbf{2}^{mathbf{3}}+ldots .+mathbf{2}^{mathbf{9}}= ) A. 1396 в. 1022 c. 1587 D. 1478 | 11 |

1057 | Assertion The number of roots of the equation ( sin left(2^{x}right) cos left(2^{x}right)=frac{1}{4}left(2^{x}+2^{-x}right) ) is 2 Reason ( A M geq G M ) for any two positive numbers. A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Assertion is incorrect but Reason is correct | 11 |

1058 | There are two A.P.’s each of ( n ) terms ( boldsymbol{a}, boldsymbol{a}+boldsymbol{d}, boldsymbol{a}+boldsymbol{2} boldsymbol{d}, ldots boldsymbol{L}, boldsymbol{p}, boldsymbol{p}+boldsymbol{q}, boldsymbol{p}+ ) ( 2 q, ldots . . L^{prime} . ) These A.P.’s satisfy the following conditions: ( frac{boldsymbol{L}}{boldsymbol{p}}=frac{boldsymbol{L}^{prime}}{boldsymbol{a}}= ) ( 4, frac{S_{n}}{S_{n^{prime}}}=2 ; ) find out ( 2left(frac{L}{L^{prime}}right) ) | 11 |

1059 | Find ( s_{30} ) for an AP, where ( a_{10}=20 ) and ( boldsymbol{a}_{20}=mathbf{5 8} ) | 11 |

1060 | The coefficient of ( x^{49} ) in the product ( (x-1)(x-3) ldots(x-99) ) is A . ( -99^{2} ) B. c. -2500 D. none of these | 11 |

1061 | The minimum value of ( boldsymbol{f}(boldsymbol{x})= ) ( 2^{log _{8}^{3} cos ^{2} x}+3^{log _{8}^{2} sin ^{2} x} ) A ( cdot 2^{1-log _{8} sqrt{3}} ) B ( cdot 2^{log _{8} sqrt{3}} ) c. ( 3^{log _{8} sqrt{2}} ) D. ( 2^{1+log _{8}} sqrt{3} ) | 11 |

1062 | The sum of 11 terms of an A.P. Whose middle term is 30 is ( ? ) A. 320 в. 330 ( c cdot 340 ) D. 350 | 11 |

1063 | If ( A, B, C ) are in A.P., then ( frac{sin A-sin C}{cos C-cos A}=? ) | 11 |

1064 | Sum of 20 terms of ( 3+6+12+ldots ) is A ( cdot 3left(2^{20}-1right) / 2 ) B . ( 3left(2^{19}-1right) / 2 ) c. ( 3left(2^{20}-1right) ) D. ( 3left(2^{19}-1right) ) | 11 |

1065 | If the mean of ( x+4,5,3,6,7 ) is 5 then find ( x ) | 11 |

1066 | If ( t_{5}, t_{10} ) and ( t_{25} ) are ( 5^{t h}, 10^{t h} ) and ( 25^{t h} ) terms of an AP respectively, then the value of ( left|begin{array}{ccc}boldsymbol{t}_{5} & boldsymbol{t}_{10} & boldsymbol{t}_{25} \ mathbf{5} & mathbf{1 0} & mathbf{2 5} \ mathbf{1} & mathbf{1} & mathbf{1}end{array}right| ) is A . -40 B. ( c cdot-1 ) D. 0 E. 40 | 11 |

1067 | Find the Odd one among : 517,661,814 922,1066,1256 A. 66 B. 814 ( c . ) 1256 D. 922 | 11 |

1068 | The common ratio of the geometric sequence ( a_{n}=3^{n-1} ) is ( A ) B. 2 ( c cdot 3 ) D. 4 | 11 |

1069 | If ( boldsymbol{S}_{boldsymbol{n}}=frac{boldsymbol{3}}{boldsymbol{4}}+frac{boldsymbol{5}}{boldsymbol{3} boldsymbol{6}}+frac{boldsymbol{7}}{boldsymbol{1 4 4}}+frac{boldsymbol{9}}{boldsymbol{4 0 0}}+ldots ) to ( boldsymbol{n} ) terms, then find ( frac{1}{1-S_{40}} ) | 11 |

1070 | For what value of ( n ), are the ( n^{t h} ) terms of two ( A . P^{prime} S 63,65,67, dots ) and ( mathbf{3}, mathbf{1 0}, mathbf{1 7} ldots . ) are equal | 11 |

1071 | Find ( 8 t h ) and 12 th terms of the G.P. ( 81,-27,9, dots dots dots ) | 11 |

1072 | The product ( 2^{frac{1}{4}} cdot 4^{frac{1}{16}} cdot 8^{frac{1}{48}} cdot 16^{frac{1}{128}} cdot ldots ) to ( infty ) is equal to? A .2 B. ( 2^{frac{1}{2}} ) c. 1 D. ( 2^{frac{1}{4}} ) | 11 |

1073 | Find the number of triplets of integers in arithmetic progression, the sum of whose squares is 1994 A . 36 B. 45 c. 12 D. Does not exist | 11 |

1074 | Mean of the first ( n ) terms of the A.P. ( boldsymbol{a},(boldsymbol{a}+boldsymbol{d}),(boldsymbol{a}+boldsymbol{2} boldsymbol{d}), ldots ldots . . ) is A ( cdot a+frac{n d}{2} ) в. ( a+frac{(n-1) d}{2} ) c. ( a+(n-1) d ) D. ( a+n d ) | 11 |

1075 | Find the 12 th term from the last term of the A.P. ( 2,6,10 ldots . .58 ) A . 110 B. 102 ( c cdot 160 ) D. 120 | 11 |

1076 | Given the terms ( a_{10}=frac{3}{512} ) and ( a_{15}= ) ( frac{3}{16384} ) of a geometric sequence, find the exact value of the term ( a_{30} ) of the sequence. | 11 |

1077 | Given a sequence of 4 members, first three of which are in G.P. and the last three are in A.P. with common difference six. If first and last terms of this sequence are equal, then the last term is: ( A cdot 8 ) B . 16 ( c cdot 2 ) D. 4 | 11 |

1078 | 4 0 , U2, U 0 . mangle, the lengths of the two larger sides are 10 and 9 espectively. If the angles are in AP. Then the length of the (1987 – 2 Marks) third side can be (a) 5-6 (c) 5 (b) 373 (d) 5+V6 (e) none | 11 |

1079 | Which term of the A.P 5,9,13,17…. is 81? | 11 |

1080 | Find the 5 th term from the end of the ( A P 7,10,13, . .154 ) | 11 |

1081 | The 4 th term of a G.P is square of its second term, and the first term is -3 Determine its 7 th term. | 11 |

1082 | Which of the following are APs ? If they form an AP, find the common difference ( d ) and write three more terms. (i) ( 2,4,8,16, dots ) (ii) ( 2, frac{5}{2}, 3, frac{7}{2}, dots ) (iii) ( -1.2,-3.2,-5.2,-7.2, dots ) ( (i v)-10,-6,-2,2, dots ) ( (v) 3,3+sqrt{2}, 3+2 sqrt{2}, 3+3 sqrt{2} ) (vi) ( 0.2,0.22,0.222,0.2222, ldots ) ( (v i i) 0,-4,-8,-12, dots ) ( (text { viii })-frac{1}{2},-frac{1}{2},-frac{1}{2},-frac{1}{2}, dots ) ( (i x) 1,3,9,27 ) ( (x) a, 2 a, 3 a, 4 a, dots ) ( (x i) a, a^{2}, a^{3}, a^{4}, dots ) ( (x i i) sqrt{2}, sqrt{8}, sqrt{18}, sqrt{32}, dots ) ( (x text { iii }) sqrt{3}, sqrt{6}, sqrt{9}, sqrt{12}, dots ) ( (x i v) 1^{2}, 3^{2}, 5^{2}, 7^{2}, . . ) ( (x v) 1^{2}, 5^{2}, 7^{2}, 73, dots ) | 11 |

1083 | 26. If a, b, c are positive real numbers. Then prove that (a +1)?(6+1)?(C+1)? >77a4544 (2004 – 4 Marks) | 11 |

1084 | Find four number forming a ( G . P . ) in which the third term is greater than the first by ( 9, ) and the second term is greater than the fourth by ( 18 . ) Write the largest of the four numbers. | 11 |

1085 | Fifth term of ( frac{1}{16}, frac{1}{8}, frac{1}{4} dots dots ) is A ( cdot frac{1}{2} ) в. ( c cdot c ) ( D ) | 11 |

1086 | Evaluate ( 1.6+2.9+3.12+ldots+ ) ( mathbf{n}(mathbf{3 n}+mathbf{3})= ) A ( cdot n(n+1)(n+2) ) B. ( (n+1)(n+2)(n+3) ) c. ( (n+2)(n+3)(n+4) ) D. ( (n-1) n(n+1) ) | 11 |

1087 | If the sum of first ( p ) terms of an ( A . P . ) is equal to the sum of first ( (p+q) ) terms is zero. Where ( boldsymbol{p} neq boldsymbol{q} ) | 11 |

1088 | Prove: ( 1^{2}+2^{2}+3^{2}+ldots ldots ldots n^{2}>frac{n^{3}}{3} ) | 11 |

1089 | The fifth term of an A.P is 1 whereas its 31st term is ( -77 . ) Find sum of its first fifteen terms. Also find which term of the series will be -17 and sum of how many terms will be 20 | 11 |

1090 | If the sequence ( a_{1}, a_{2}, a_{3}, dots ) is in A.P. then the sequence ( a_{5}, a_{10}, a_{15}, dots ) is ( A ). A G.P. B. An A.P. c. Neither A.P. nor G.P. D. A constant sequence | 11 |

1091 | If ( log _{10} 2, log _{10}left(2^{x}-1right) ) and ( log _{10}left(2^{x}+3right) ) be three consecutive terms of an A.P., then ( mathbf{A} cdot x=0 ) B. ( x=1 ) C ( . x=log _{2} 5 ) D. ( x=log _{10} 2 ) | 11 |

1092 | Find the arithmetic mean of the progression 2,4,6,8,10 A . 10 B . 20 c. 30 ( D ) | 11 |

1093 | For ( x>1, ) the least value of the expression ( 2 log _{10} x-log _{x}(0 cdot 01) ) is : A . 10 B. -0.01 ( c cdot 2 ) D. None of these | 11 |

1094 | An example of G.P. is A ( cdot-1, frac{1}{2}, frac{1}{4}, frac{1}{8} dots ) в. ( -1, frac{3}{2}, frac{1}{2},-frac{1}{2} ) c. ( _{1, frac{1}{2}, frac{1}{4}, frac{1}{6} ldots} ) D. ( 1, frac{1}{2}, frac{1}{4}, frac{1}{8} ldots ) | 11 |

1095 | An AP consists of 37 terms. The sum of the three middle most terms is 225 and the sum of the last three is ( 429 . ) Find the AP. | 11 |

1096 | ( 1^{2}, 5^{2}, 7^{2}, 73 dots dots dots ) is it an AP? If yes, then what is it’s common difference? A. No B. Yes, ( d=15 ) c. Yes, ( d=24 ) D. Yes, ( d=25 ) | 11 |

1097 | Let ( p, q, r epsilon R^{+} ) and ( 27 p q r geq(p+q+r)^{3} ) and ( 3 p+4 q+5 r=12 ) then ( p^{3}+q^{4}+ ) ( r^{5} ) is equal to ( A cdot 3 ) B. 6 ( c cdot 2 ) ( D ) | 11 |

1098 | The sum of infinite series ( frac{1.3}{2}+frac{3.5}{2^{2}}+ ) ( frac{5.7}{2^{3}}+frac{7.9}{2^{4}}+ldots infty ) A . 21 B . 22 ( c cdot 23 ) D. None | 11 |

1099 | The sequence ( 6,12,24,48 dots ) is a A . geometric series B. arithmetic sequence c. geometric progression D. harmonic sequence | 11 |

1100 | Find ( a_{n} ) in an ( A P ) if ( a=1, d=1, n=3 ) | 11 |

1101 | What is the common ratio of the geometric sequence ( 81,27,9,3, dots ? ) A ( cdot frac{1}{2} ) B. ( c cdot frac{1}{4} ) D. | 11 |

1102 | If the mean of 5 observations ( x, x+ ) ( 2, x+4, x+6 ) and ( x+8 ) is ( 11, ) find the value of ( x ) | 11 |

1103 | ( mathrm{n} triangle boldsymbol{A B C}, sumleft(frac{sin ^{2} boldsymbol{A}+sin boldsymbol{A}+mathbf{1}}{sin boldsymbol{A}}right) ) is always greater than ( A cdot 9 ) B. 3 c. 27 D. None of these | 11 |

1104 | The sum of three terms of a geometric sequence is ( frac{39}{10} ) and their product is 1 Find the common ratio and the terms | 11 |

1105 | If the sum of ( n ) terms of a GP (with common ratio ( r ) ) beginning with the ( p^{t h} ) term is ( k ) times the sum of an equal number of the same series beginning with the ( q^{t h} ) term, then the value of ( k ) is A ( cdot r^{p / q} ) B ( cdot r^{a / p} ) c. ( r^{p-q} ) D. ( r^{p+q} ) | 11 |

1106 | The inventor of the chess board suggested a reward of one gram of wheat for the first square, 2 grains for the second, 4 grains for the third and so on, doubling the number of the grains subsequent squares. How many grins would to be given to inventor? ( There are 64 squares in the chess board) | 11 |

1107 | The sum of ‘n’ terms of two A.P.’s are in the ratio of ( frac{5 n+2}{11 n-7} . ) Find the ratio of their sixth terms. A . 32: 59 B. 1: 1 c. 2: 1 D. 1: 2 | 11 |

1108 | If ( S_{n}=n^{2} p ) and ( S_{m}=m^{2} p, m neq n, ) in an A.P., then ( S_{p}=p^{3} ) A. True B. False | 11 |

1109 | State the whether given statement is true or false If the ( 9^{t h} ) term of an ( A . P . ) is zero, then | 11 |

1110 | If ( boldsymbol{x}, boldsymbol{y}, boldsymbol{z} ) are in A.P., then ( boldsymbol{2} boldsymbol{y}= ) ( mathbf{A} cdot x+z ) B. ( x-z ) c. ( sqrt{x y} ) D. ( x z ) | 11 |

1111 | The number of terms in a sequence ( 6,12,24, dots .1536 ) represents a A . arithmetic progression B. harmonic progression c. geometric progression D. geometric series | 11 |

1112 | What is ( a_{4} ) when ( a_{1}=2, r=-3 ? ) A .27 в. -27 ( c .-54 ) D. 54 | 11 |

1113 | There are 37 terms in an ( A . P ., ) the sum of three terms placed exactly at the middle is 225 and the sum of last three terms is ( 429 . ) Write the ( A . P ) | 11 |

1114 | The eight term of an ( A . P . ) is half the second term and eleventh term exceeds one third of its fourth term by 1. Find its ( 15^{t h} ) term | 11 |

1115 | There exist an (infinite) non-constant arithmetical progression whose terms are all prime numbers If true then enter 1 and if false then enter 0 | 11 |

1116 | Insert two arithmetic means between 11 and 17 A. 13 and 15 B. 11 and 13 c. 13 and 14 D. 11 and 15 | 11 |

1117 | Find the ( 10^{t h} ) term ( & ) nth term of the G.P. 5,25,125 | 11 |

1118 | Prove that: ( boldsymbol{a}_{n^{2}+1}=left(boldsymbol{n}^{2}+mathbf{1}^{2}right)-left(boldsymbol{n}^{2}+mathbf{1}right)+mathbf{1}= ) ( left(n^{2}+n+1right)left(n^{2}-n+1right)= ) ( mathbf{A} cdot a_{n-1} a_{n} ) В ( cdot a_{n+1} a_{n} ) ( mathbf{c} cdot a_{n+1} a_{n-1} ) D. ( a_{n+2} a_{n} ) | 11 |

1119 | Determine the A.P. whose fifth tern is 19 and different one of the eight term from the thirteenth term is 20 | 11 |

1120 | Find the number of terms in each of the following AP’s ( 7,13,19, dots 205 ) A. 20 terms B. 28 terms c. 34 terms D. 40 terms | 11 |

1121 | ( 1+3+6+10+ldots ) upto ( n ) terms is equal to A ( cdot frac{1}{3} n(n+1)(n+2) ) B ( cdot frac{1}{6} n(n+1)(n+2) ) c. ( frac{1}{12} n(n+2)(n+3) ) D ( cdot frac{1}{12} n(n+1)(n+2) ) | 11 |

1122 | If ( boldsymbol{a}_{1} boldsymbol{a}_{2} ldots ldots ldots ldots boldsymbol{a}_{boldsymbol{n}} boldsymbol{epsilon} boldsymbol{R}^{+} boldsymbol{a} boldsymbol{n} boldsymbol{d} boldsymbol{a}_{1} boldsymbol{a}_{2} ldots ldots boldsymbol{a}_{n}= ) 1 then the least value of ( left(1+a_{1}+right. ) ( left.boldsymbol{a}_{1}^{2}right)left(mathbf{1}+mathbf{1}_{2}+boldsymbol{a}_{2}^{2}right) ldots ldots ldotsleft(1+boldsymbol{a}_{n}+boldsymbol{a}_{n}^{2}right) ) is ( mathbf{A} cdot 3^{n} ) В . ( n 3^{n} ) ( c cdot 3^{3 n} ) D. Data inadequate | 11 |

1123 | If the ( n^{t h} ) term of an A.P. is ( 4 n+1 ), then the common difference, ( d ) is equal to A . 3 B. 4 ( c .5 ) D. 6 | 11 |

1124 | What is the sum of G.P. ( 1,3,9,27, dots . . ) up to 7 numbers? A. 1093 в. 2093 ( c .3093 ) D. 4093 | 11 |

1125 | Write the ( n^{t h} ) term ( a_{n} ) of the ( A P ) with first term ( a ) and common difference ( d ) | 11 |

1126 | In which of the following situations, the sequence formed will form an A.P? 1) Number of students left in the school auditorium from the total strength of 1000 students when they leave the auditorium in batches of 25 2) The amount of money in the account every year when Rs 100 are deposited annually to accumulate at compound interest at ( 4 % ) per annum. | 11 |

1127 | Find the ( A M, G M ) and ( H M ) between 12 and 30 | 11 |

1128 | State whether the given statement is true or false: -150 is a term of A.P. ( 17,12,7,2, dots dots dots ) A. True B. False | 11 |

1129 | Find the sum to infinity of the geometric series ( 2+frac{2}{3}+frac{2}{9}+dots ) | 11 |

1130 | Find ( a, b ) such that ( 18, a, b,-3 ) are in ( boldsymbol{A P} ) | 11 |

1131 | A man starts repaying a loan as first installment of Rs. ( 100 . ) If he increases the installment by Rs 5 every month. what amount he will pay in the ( 30^{t h} ) installment? | 11 |

1132 | If ( frac{a-b}{b-c}=frac{a}{a}, ) then ( a, b, c ) are in ( A cdot G P ) в. нР ( c cdot A P ) D. sp | 11 |

1133 | If in a ( G P ) the ( (p+q)^{t h} ) term is ( a ) and the ( (p-q)^{t h} ) term is ( b, ) then ( p^{t h} ) term is ( mathbf{A} cdot(a b)^{1 / 3} ) B ( cdot(a b)^{1 / 2} ) ( mathbf{c} cdot(a b)^{1 / 4} ) D. None of these | 11 |

1134 | For all positive value of ( x ) and ( y, ) the value of ( frac{left(1+x+x^{2}right)left(1+y+y^{2}right)}{x y} ) is ( A . leq 9 ) в. ( 9 ) | 11 |

1135 | Which term of the GP ( 2,8,32, dots ) is ( mathbf{1 3 1 0 7 2} ? ) ( mathbf{A} cdot 6^{t h} ) B. ( 7^{text {th }} ) ( c cdot 8^{t h} ) ( mathbf{D} cdot 9^{t h} ) | 11 |

1136 | The geometric mean between two numbers ( a ) and ( b ) is A. ( sqrt{a b} ) B. ( frac{1}{sqrt{a b}} ) c. ( sqrt{2 a b} ) D. ( 2 sqrt{a b} ) | 11 |

1137 | If for an A.P. ( S_{16}=784, a=4 ) find ( d=? ) | 11 |

1138 | If the ( (boldsymbol{m}+mathbf{1}) ) th ( ,(boldsymbol{n}+mathbf{1}) ) th and ( (boldsymbol{r}+mathbf{1}) ) th terms of an A.P are in G.P and ( m, n, r ) are in H.P, then the ratio of the first term of the A.P to its common difference in terms of ( n ) be ( n / k ).Find ( -k ? ) | 11 |

1139 | ( ln ) an A.P. the first term is 2 , the last term ( 29, ) the sum ( 155 ; ) find the difference. | 11 |

1140 | The sum of ‘ ( n^{prime} ) terms of series [ left(1-frac{1}{n}right)+left(1-frac{2}{n}right)+left(1-frac{3}{n}right)+ ] will be: | 11 |

1141 | Find the term ( t_{10} ) of an A.P. ( 4,9,14, dots ) | 11 |

1142 | The geometric mean of three numbers 3,5,7 is A. 4.717 B. 3.717 c. 2.717 D. 1.717 | 11 |

1143 | If ( a, b, c ) are in A.P. ( b-a, c-b ) and ( a ) in G.P., then ( boldsymbol{a}: boldsymbol{b}: boldsymbol{c} ) is A . 1: 2: 3 B. 1: 3: 5 ( c cdot 2: 3: 4 ) D. None of these | 11 |

1144 | find ( 1+frac{1}{1+2}+frac{1}{1+2+3}+ ) to n terms. | 11 |

1145 | The sides of a right angled triangle are in A.P. The ratio of sides is A .1: 2: 3 B. 2:3:4 ( c cdot 3: 4: 5 ) ( D cdot 5: 8: 3 ) | 11 |

1146 | Solve: ( 2-frac{3}{5} ) | 11 |

1147 | ( mathbf{1 . 4}+mathbf{2 . 5}+ldots+mathbf{n}(mathbf{n}+mathbf{3})= ) A. ( frac{n(n+3)(n+5)}{9} ) в. ( frac{n(n+1)(n+5)}{3} ) c. ( frac{n(n+5)(n+7)}{6} ) D. ( frac{n(n+3)(n+9)}{12} ) | 11 |

1148 | The first, second and last terms of an A.P. are ( alpha, beta, gamma ) respectively then the sum of first ( n ) terms is A. ( beta+gamma-2 alpha ) B. ( frac{beta+gamma-2 alpha}{beta-alpha} ) C. ( frac{beta+gamma+2 alpha}{beta+alpha} ) D. ( frac{(alpha+gamma)(beta+gamma-2 alpha)}{2(beta-alpha)} ) | 11 |

1149 | If the sum of three numbers in A.P., is 24 and their product is ( 440, ) find the numbers. | 11 |

1150 | If ( S_{n}=n^{2} p ) and ( S_{m}=m^{2} p, m neq n, ) in an A.P, prove that ( boldsymbol{S}_{boldsymbol{p}}=boldsymbol{p}^{boldsymbol{3}} ) | 11 |

1151 | If ( frac{1}{boldsymbol{q}+boldsymbol{r}}, frac{mathbf{1}}{boldsymbol{r}+boldsymbol{p}} ) and ( frac{mathbf{1}}{boldsymbol{p}+boldsymbol{q}} ) are in AP then ( p^{2}, q^{2} ) and ( r^{2} ) are in ( A cdot A P ) в. GP c. нр D. AGP | 11 |

1152 | Find out the general form of geometric progression. A .2,4,8,16 в. 2,-2,2,3,1 c. 0,3,6,9,12 D. 10,20,30,40 | 11 |

1153 | Let ( S_{1}, S_{2}, dots . S_{n} ) be squares such that for each ( n geq 1, ) the length of a side of ( S_{n} ) equals the length of the diagonal of ( S_{n+1} . ) If the length of a side of ( S_{1} ) is ( 10 mathrm{cm}, ) then the least value of ( n ) for which the area of ( S_{n} ) less than 1 sq ( c m ) A. 7 B. 8 c. 9 D. 10 | 11 |

1154 | Prove that; ( a^{2}-b^{2}, b^{2}-c^{2}, c^{2}-d^{2} ) are in ( G . P ) | 11 |

1155 | GM of the numbers ( 3,3^{2}, 3^{3}, ldots . ., 3^{n} ) is. ( A cdot 3^{2 / n} ) B. ( 3^{n / 2} ) ( c cdot 3^{(n+1) / 2} ) D. ( 3^{(n-1) / 2} ) | 11 |

1156 | If ( a, b, c ) are three positive numbers,then the minimum value of the expression ( frac{boldsymbol{a b}(boldsymbol{a}+boldsymbol{b})+boldsymbol{b} boldsymbol{c}(boldsymbol{b}+boldsymbol{c})+boldsymbol{c} boldsymbol{a}(boldsymbol{c}+boldsymbol{a})}{boldsymbol{b} boldsymbol{c} boldsymbol{a}} ) A . 3 B. 4 ( c .6 ) ( D ) | 11 |

1157 | If ( S_{1}, S_{2} ) and ( S_{3} ). are the sums of first ( n ) natural numbers, their squares and their cubes respectively, then ( boldsymbol{S}_{3}left(mathbf{1}+mathbf{8} boldsymbol{S}_{1}right)= ) ( A cdot S_{2}^{2} ) в. ( 9 S_{2} ) ( mathrm{c} cdot 9 S_{2}^{2} ) D. None | 11 |

1158 | The sum of the first tern and the fifth term of an ascending A.P. is 26 and the product. of the second term by the fourth term is ( 160 . ) And the sum of the first seven terms of this AP. A . 110 B. 114 c. 112 D. 116 | 11 |

1159 | An arithmetic progression consists of 12 terms whose sum is 354 .The ratio of the sum of the even terms to the sum of the odd terms is ( 32: 27 . ) Find the common difference of the progression. | 11 |

1160 | Find out whether the sequence ( 1^{2}, 3^{2}, 5^{2}, 7^{2}, dots ) is an AP. If it is, find out the common difference. ( A cdot ) No B. Yes, ( d=8 ) ( mathbf{c} . ) Yes, ( d=-8 ) D. Yes, ( d=9 ) | 11 |

1161 | If ( a, b, c ) and ( d ) are in ( G . P . ) show that ( left(a^{2}+b^{2}+c^{2}right)left(b^{2}+c^{2}+d^{2}right)= ) ( (a b+b c+c d)^{2} ) | 11 |

1162 | If ( boldsymbol{a}_{mathbf{1}} in boldsymbol{R}-{mathbf{0}}, boldsymbol{i}=mathbf{1}, mathbf{2}, mathbf{3}, boldsymbol{4} ) and ( boldsymbol{x} in boldsymbol{R} ) and ( left(sum_{i=1}^{3} a_{i}^{2}right) x^{2}- ) ( mathbf{2} boldsymbol{x}left(sum_{i=1}^{3} boldsymbol{a}_{i} boldsymbol{a}_{i+1}right)+sum_{i=2}^{4} boldsymbol{a}_{i}^{2} leq mathbf{0}, ) then ( a_{1}, a_{2}, a_{3}, a_{4} ) are in A . A. в. G. c. н. D. A.G. | 11 |

1163 | If the ( 3 r d ) and the ( 9 t h ) terms of an ( A P ) are 4 and -8 respectively, which term of this is zero? | 11 |

1164 | Sum to infinite terms the following series: ( mathbf{1}+mathbf{3} boldsymbol{x}+mathbf{5} boldsymbol{x}^{2}+mathbf{7} boldsymbol{x}^{3}+ldots .,|boldsymbol{x}|<mathbf{1} ) | 11 |

1165 | If for a G.P. ( S_{6}=126 ) and ( S_{3}=14 ) then find ( mathbf{r} ) A B. 2 ( c cdot 3 ) D. 4 | 11 |

1166 | If ( s ) and ( t ) respectively the sum and the sum of the squares of n successive positive integers beginning with a, then show that ( n t-s^{2} ) is independent of a. | 11 |

1167 | A geometric sequence can be written as A ( cdot a r, a r^{2}, a r^{4}, a r^{6} dots a r^{n} ) B . ( a, a r^{2}, a r^{3}, a r^{4} ldots a r^{n} ) C. ( a, a r, a r^{1}, a r^{3}, a r^{4} dots a r^{n} ) D. ( a, a r, a r^{2}, a r^{3}, a r^{4} dots a r^{n} ) | 11 |

1168 | ( 1.3+3.5+5.7+ldots+(2 n-1)(2 n+1)= ) ( frac{nleft(4 n^{2}+6 n-1right)}{3} ) | 11 |

1169 | Which term of an AP: 2, -1, – 4, …. is ( -70 ? ) A . 15 th B. 18th c. 25th D. 30th | 11 |

1170 | The sum of ( n ) terms of the series ( 1^{2}- ) ( mathbf{2}^{2}+mathbf{3}^{2}-mathbf{4}^{2}+mathbf{5}^{2}-mathbf{6}^{2}+ldots ) is This question has multiple correct options A ( cdot frac{-n(n+1)}{2} ) if ( n ) is ever B. ( frac{n(n+1)}{2} ) if ( n ) is od c. ( -n(n+1) ) if ( n ) is even D. ( frac{n(n+1)(2 n+1)}{6} ) if ( n ) is odd | 11 |

1171 | Prove ( frac{b c}{a}+frac{c a}{b}+frac{a b}{c} geq a+b+c, ) for ( a, b, c>0 ) | 11 |

1172 | The first terms of a G.P. is 1. The sum of the third and fifth term is ( 90 . ) Find the common ratio. | 11 |

1173 | State true or false. ( mathbf{1 9 9 . 1}+mathbf{1 9 7} . mathbf{3}+mathbf{1 9 5 . 5}+ldots . mathbf{3 . 1 9 7}+ ) ( 1.397=666700 ) A. True B. False | 11 |

1174 | If ( frac{2}{3}, k, frac{5 k}{8} ) are in A.P. then ( mathbf{k}= ) | 11 |

1175 | If ( 7 t h ) and ( 13 t h ) terms of an ( A . P . ) Be 34 and ( 64, ) respectively, then its 18 th terms is: ( A cdot 87 ) B. 88 c. 89 D. 90 | 11 |

1176 | Given ( a=7, a_{13}=35, ) find the d A . 2.33 B. 2.72 ( c . ) 3.89 D. 4.56 | 11 |

1177 | 7th term of ( frac{1}{2}, frac{1}{4}, frac{1}{6} dots dots ) is ( A cdot frac{1}{10} ) в. ( frac{1}{12} ) c. ( frac{1}{14} ) D. ( frac{1}{16} ) | 11 |

1178 | The sum of 10 terms of the series ( 0.7+ ) ( .77+.777+ldots ldots ldots ) is A ( cdot frac{7}{9}left(89+frac{1}{10^{10}}right) ) в. ( frac{7}{81}left(89+frac{1}{10^{10}}right) ) c. ( frac{7}{81}left(89+frac{1}{10^{9}}right) ) D. ( frac{7}{9}left(89+frac{1}{10^{9}}right) ) | 11 |

1179 | 87,78,69 Write the next number of the series. A . 26 B. 60 c. 12 D. zero | 11 |

1180 | In set of 4 number the first three number are in ( G P ) and the last three are in ( A P ) with common different is 6 If first number is same as the fourth number. Find the numbers | 11 |

1181 | If ( a, b, c ) are in A.P., then ( frac{1}{sqrt{b}+sqrt{c}}, frac{1}{sqrt{c}+sqrt{a}}, frac{1}{sqrt{a}+sqrt{b}} ) are in A. ( A . P ) в. G.Р. c. ( H . P ) D. None of these | 11 |

1182 | Prove that the sum of ( n ) number of terms of two different A.P.’s can be same for only one value of ( n ) if they have same value of ( boldsymbol{d}, boldsymbol{a} ) | 11 |

1183 | Find the sum of even numbers between and 25 A. 155 в. 156 c. 157 D. 158 | 11 |

1184 | The value of ( x ) that satisfies the relation ( boldsymbol{x}=mathbf{1}-boldsymbol{x}+boldsymbol{x}^{2}-boldsymbol{x}^{mathbf{3}}+boldsymbol{x}^{4}-boldsymbol{x}^{mathbf{5}}+. infty ) if ( |boldsymbol{x}|<mathbf{1} ) A ( cdot frac{-1 pm sqrt{5}}{2} ) B. ( frac{-1 pm 3 i}{2} ) ( c ) D. none | 11 |

1185 | Expansion of series: ( sum_{n=0}^{4} 2 n ) A. ( 0+2+4+8+16 ) B. ( 0+2+4+6+8 ) c. ( 2+4+6+8+10 ) D. None of the above | 11 |

1186 | If ( a, b, c ) are positive such that ( a b^{2} c^{3}= ) 64 then least value of ( left(frac{1}{a}+frac{2}{b}+frac{3}{c}right) ) is ( A cdot 6 ) B. 2 ( c cdot 3 ) D. 32 | 11 |

1187 | The ( n^{t h} ) term of an ( A P ) is ( a=2=n+1 ) find its sum. | 11 |

1188 | If ( x_{1}, x_{2}, dots ., x_{n} ) are any real number and ( boldsymbol{n} ) is any positive integer, then ( ? ) B ( cdot n sum_{i=1}^{n} x_{i}^{2} geqleft(sum_{i=1}^{n} x_{i}right)^{2} ) c. ( n sum_{i=1}^{n} x_{i}^{2} geq nleft(sum_{i=1}^{n} x_{i}right)^{2} ) D. None of the above | 11 |

1189 | Find ( A M ) of divisors of 100 . ( mathbf{A} cdot 24 ) B . 25. c. 24.11 D. 21.9 | 11 |

1190 | ( a b c d=81 ) find minimum value of ( a+ ) ( boldsymbol{b}+boldsymbol{c}+boldsymbol{d} ? ) A . 12 B. ( c cdot-1 ) D. -12 | 11 |

1191 | The first and the last terms of an A.P are 5 and 45 respectively. If the sum of all its terms is ( 400, ) find its common difference. A. ( _{d=frac{5}{3}} ) B. ( d=frac{8}{5} ) c. ( _{d=frac{8}{3}} ) D. None of these | 11 |

1192 | Which term of the ( boldsymbol{A P}: 88,84,80, ldots ldots ) is zero? | 11 |

1193 | Find the common difference of the A.P. and write the next two terms: ( mathbf{7 5}, mathbf{6 7}, mathbf{5 9}, mathbf{5 1}, dots . . ) | 11 |

1194 | If the third and ( 11^{t h} ) term of an A.P are 8 and 20 respectively, find the sum of first ten terms. A ( cdot 105 frac{1}{2} ) в. 108 c. ( _{117} frac{1}{2} ) D. ( 203 frac{1}{2} ) | 11 |

1195 | Sum of the series ( boldsymbol{S}=mathbf{1}+frac{mathbf{1}}{mathbf{2}}(mathbf{1}+mathbf{2})+frac{mathbf{1}}{mathbf{3}}(mathbf{1}+mathbf{2}+mathbf{3})+ ) ( frac{1}{4}(1+2+3+4)+ldots ) upto 20 terms is A. 110.5 B. 111.5 c. 115.5 D. 116.5 | 11 |

1196 | Find ( A M ) of 7 and 27 | 11 |

1197 | If sum of ( n ) terms of an A.P.is given by ( boldsymbol{S}_{n}=boldsymbol{a}+boldsymbol{b} boldsymbol{n}+boldsymbol{c} boldsymbol{n}^{2} ) where ( mathbf{a}, mathbf{b}, mathbf{c} ) are independent of ( n, ) then This question has multiple correct options ( mathbf{A} cdot a=0 ) B. Common difference of A.P. must be 2b. c. Common difference of A.P. must be 2 c D. All above | 11 |

1198 | The sum of odd integers from 1 to 2001 is A ( cdot(1121)^{2} ) В. ( (1101)^{2} ) c. ( (1001)^{2} ) D. ( (1021)^{2} ) E ( cdot(1011)^{2} ) | 11 |

1199 | For an A.P.. ( mathrm{S}_{2 n}=3 S_{n} . ) The value of ( frac{S_{3 n}}{S_{n}} ) is equal to A . 4 B. 6 c. 8 D. 10 | 11 |

1200 | The nth term of a series is given to be ( frac{3+n}{4}, ) find the sum of 105 term of this series. | 11 |

1201 | suppose a, a,, ….,a,,, are in A.P. such that b log b.. …., loge 19. Let b;> 1 for i= 1, 2, …, 101. Suppose log, b, logeb2, .., 6101 are in Arithmetic Progression (A.P.) with the com difference log, 2. Suppose a, a,, …., a 101 ar a, b, and as,= be. Ift=b,+b, + …. + be, and s=a, +a+…+ a53, then (JEE Adv. 2016) (a) s>t and a., > b (b) s>t and a 101<5101 (c) s bo (d) s<t and a on <b101 | 11 |

1202 | A person opens an account with 50 and starts depositing every day double the amount he has deposited on the previous day. Then find the amount he has deposited on the 10 th day from the beginning. A . 25000 B. 25600 c. 28500 D. 26500 | 11 |

1203 | (2002 – Marks) 17. If a, b, c are in A.P., a2, 62, c2 are in H.P., then prove that either a=b=c or a, b, – form a G. (2003 – 4 Marks) | 11 |

1204 | If the sums of ( p, q ) and ( r ) terms of an A.P. be ( a, b ) and ( c ) respectively, then prove ( operatorname{that} frac{boldsymbol{a}}{boldsymbol{p}}(boldsymbol{q}-boldsymbol{r})+frac{boldsymbol{b}}{boldsymbol{q}}(boldsymbol{r}-boldsymbol{p})+frac{boldsymbol{c}}{boldsymbol{r}}(boldsymbol{p}- ) ( boldsymbol{q})=mathbf{0} ) | 11 |

1205 | , then find the value of 8 Illustration 3.22 If A = tan (rA) tan((r +1)A). r=1 | 11 |

1206 | The sum of the first ( n ) terms of the geometric progression, whose first term is 4 and the common ratio is ( 3, ) is 4372 Find ( n ) | 11 |

1207 | The arithmetic mean of 12 and 20 is : A . 12 B. 14 c. 16 D. 18 | 11 |

1208 | If ( mathbf{x} in mathbf{R} ) and the numbers ( mathbf{5}^{(1-x)}+mathbf{5}^{(x+1)} ) a/2, ( left(25^{x}+25^{-x}right) ) form an A. P. then a must lie in the interval ( A cdot[12, infty] ) в. [-5,5] D. ( [8, infty] ) | 11 |

1209 | Three numbers whose sum is 45 are in A.P. If 5 is subtracted from the first number and 25 is added to third number, the numbers are in G.P. Then numbers can be A. 10,15,20 B. 8,15,22 c. 5,15,25 D. 12,15,18 | 11 |

1210 | The list of numbers ( 10,6,2,-2, cdots ) is A . an A.P. with ( d=16 ) B. an A.P. with ( d=-4 ) c. an ( A . P . ) with ( d=4 ) D. not an ( A . P ). | 11 |

1211 | The sum of the infinite series ( frac{1.3}{2}+ ) ( frac{3.5}{2^{2}}+frac{5.7}{2^{3}}+frac{7.9}{2^{4}}+dots dots dots . .0 . ) | 11 |

1212 | The algebraic form of an arithmetic sequence is ( 5 n+3 ) a. What is the first form of the sequence? b. What will be the remainder if the term of the sequences are divide by ( 5 ? ) | 11 |

1213 | If ( T_{n}=3 n+8, ) then ( T_{n-1}= ) A. ( 3 n+7 ) B. ( 3 n+6 ) c. ( 3 n-5 ) D. ( 3 n+5 ) | 11 |

1214 | Find ( 4^{t h} ) term from the end of the G.P. ( mathbf{3}, mathbf{6}, mathbf{1 2}, mathbf{2 4}, dots, mathbf{3 0 7 2} ) | 11 |

1215 | What is the average of the first 300 terms of the given sequence? ( mathbf{1},-mathbf{2}, mathbf{3},-mathbf{4}, mathbf{5},-mathbf{6}, dots, dots, mathbf{n} .(-mathbf{1})^{n+1} ) A . -1 в. 0.5 ( c cdot 0 ) D. – 0.5 | 11 |

1216 | The A.M. of two numbers exceeds their G.M. by 15 and H.M. by 27 , find the bigger of the two numbers. | 11 |

1217 | If ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} ) are in A.P. show that ( cot boldsymbol{beta}= ) ( frac{sin alpha-sin gamma}{cos gamma-cos alpha} ) | 11 |

1218 | Two A.P’s have the same common difference. The difference between their ( 100^{t h} ) terms is ( 111222333 . ) What is the difference between their millionth terms? A . 111222333 B. 222 111 333 c. 333 111222 D. 111333222 | 11 |

1219 | ( mathbf{f} boldsymbol{x}=sum_{n=0}^{infty} boldsymbol{a}^{n}, boldsymbol{y}=sum_{n=0}^{infty} boldsymbol{b}^{n}, boldsymbol{z}=sum_{n=0}^{infty}(boldsymbol{a} boldsymbol{b})^{n} ) where ( a, b<1 ) then prove that ( x z+ ) ( boldsymbol{y} boldsymbol{z}=boldsymbol{x} boldsymbol{y}+boldsymbol{z} ) Another form: For ( mathbf{0}<boldsymbol{theta}, boldsymbol{phi}-frac{boldsymbol{pi}}{mathbf{2}} ) if ( boldsymbol{x}=sum_{n=0}^{infty} cos ^{2 n} boldsymbol{theta} ) ( boldsymbol{y}=sum_{n=0}^{infty} sin ^{2 n} phi, z=sum_{n=0}^{infty} cos ^{2 n} theta sin ^{2 n} phi ) then prove that ( boldsymbol{x} boldsymbol{z}+boldsymbol{y} boldsymbol{z}-boldsymbol{z}=boldsymbol{x} boldsymbol{y} ) | 11 |

1220 | ( frac{1^{2}}{1}+frac{1^{2}+2^{2}}{1+2}+frac{1^{2}+2^{2}+3^{2}}{1+2+3}+ldots+n ) terms ( = ) A. ( frac{n(n+3)}{4} ) в. ( frac{n(n+3)}{5} ) c. ( frac{n(n+2)}{3} ) D. ( frac{n(n+5)}{6} ) | 11 |

1221 | Which term of the AP ( 121,117,113, ldots ldots ) is the first negative | 11 |

1222 | The common difference of an A.P. in which ( a_{25}-a_{12}=-52 ) is: A .4 B. -4 ( c cdot-3 ) D. 3 | 11 |

1223 | Assertion STATEMENT-1: ( 1.3 .5 ldots . .(2 n-1)> ) ( boldsymbol{n}^{boldsymbol{n}}, boldsymbol{n} boldsymbol{epsilon} boldsymbol{N} ) Reason STATEMENT-2: The sum of the first ( n ) odd natural numbers is equal to ( n^{2} ) A. Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement- B. Statement-1 is True, Statement-2 is True; Statementis Not a correct explanation for Statement- c. statement- – is True, Statement-2 is False D. Statement- 1 is False, Statement- 2 is True | 11 |

1224 | Solve the following equations. ( 7^{x+2}-frac{1}{7} cdot 7^{x+1}-14 cdot 7^{x-1}+2 cdot 7^{x}=49 ) | 11 |

1225 | If ( a, b, c ) are in G.P., then A ( cdot a^{2}, b^{2}, c^{2} ) are in G.P. B . ( a^{2}(b+c), c^{2}(a+b), b^{2}(a+c) ) are in G.P. c. ( frac{a}{b+c}, frac{b}{c+a}, frac{c}{a+b} ) are in G.P D. None of the above. | 11 |

1226 | State True or False. ( 1<frac{1}{1001}+frac{1}{1002}+frac{1}{1003}+dots dots+ ) ( frac{1}{3001}<1 frac{1}{3} ) A. True B. False | 11 |

1227 | Find the first three terms of an infinitely decreasing geometric progression whose sum is 1.6 and the second term is ( -mathbf{0 . 5} ) | 11 |

1228 | The number of real solutions of the equation ( sin left(e^{x}right)=2018^{x}+2018^{-x} ) is ( mathbf{A} cdot mathbf{0} ) B. ( c cdot 2 ) D. infinitely many | 11 |

1229 | Given ( a_{n}=4, d=2, S_{n}=-14, ) then find the ( n ) and ( a_{n} ) | 11 |

1230 | Let ( a, b, c ) be three distinct positive real numbers in G.P., then ( a^{2}+2 b c-3 a c ) is ( A cdot>0 ) ( B .<0 ) ( c_{c}=0 ) D. can't be found out | 11 |

1231 | The product of ( t_{5} ) and ( t_{6} ) of the progress ( frac{1}{4}, frac{1}{2}, 1, dots dots ) is A ( cdot t_{8} ) в. ( t_{11} ) c. ( t_{1} ) D. ( t ) | 11 |

1232 | Find the sum of first 32 multiples of 4 ( mathbf{A} cdot 2,112 ) B. 2,712 ( c cdot 2,110 ) D. 2,111 | 11 |

1233 | The arithmetic mean of the nine numbers in the given set ( {9,99,999, dots dots . .999999999} ) is a 9 digit number ( N, ) all whose digits are distinct, The number ( N ) does not contain the digit ( mathbf{A} cdot mathbf{0} ) B. 2 c. 5 D. | 11 |

1234 | Find the 50 th term of the sequence: ( frac{1}{n}, frac{n+1}{n}, frac{2 n+1}{n} dots dots dots dots ) | 11 |

1235 | 15. An infinite GP. has first term ‘x’ and sum ‘5’, then x belongs to (2004S) (a) x<-10 (b) -10<x<0 (c) 0<x10 | 11 |

1236 | The number of real roots of ( sin left(2^{x}right) cos left(2^{x}right)=frac{1}{4}left(2^{x}+2^{-x}right) ) is ( A ) B. 2 ( c cdot 3 ) D. No solution | 11 |

1237 | How many terms of the series ( -9,-6,-3, dots ) must be taken that the sum may be ( 66 ? ) | 11 |

1238 | In a sequence of 21 terms, the first 11 term are in ( A . P ) with common difference 2 and the last 11 terms are in ( G . P . ) with common ratio ( 2 . ) If the middle term of the ( A . P . ) is equal to the middle term of ( G . P . ) The find the middle term of the entire sequence. | 11 |

1239 | Find ( boldsymbol{n} boldsymbol{t h} ) term is the ( mathbf{A . P} mathbf{1 3}, mathbf{8}, mathbf{3}, mathbf{2}, dots . ) | 11 |

1240 | The ( 15^{t h} ) terms of the series ( 2 frac{1}{2}+ ) ( 1 frac{7}{13}+1 frac{1}{9}+frac{20}{23}+_{—-} ) ( A cdot frac{5}{12} ) в. ( frac{10}{21} ) c. ( frac{10}{39} ) D. None of these | 11 |

1241 | If ( a,(a-2) ) and ( 3 a ) are in ( A P, ) then the value of ( a ) is A . -3 B. – ( c .3 ) D. 2 | 11 |

1242 | If ( a, b, c ) are in A.P., then value of the expression ( a^{3}+c^{3}-8 b^{3} ) is equal to A . ( 2 a b c ) B. ( 6 a b c ) c. ( 4 a b c ) D. ( -6 a b c ) | 11 |

1243 | Let p and q be roots of the equation x2 – 2x + A= 0 and letr and s be the roots of the equation x2 – 18x + B = 0. If p<q<r<s are in arithmetic progression, then A= and B= ……. (1997 – 2 Marks) | 11 |

1244 | 13. Let a, b, c, d bere system of equations et ab, c, d be real numbers in G.P. Ifu, v, w, satisfy the (1999 – 10 Marks) u +2v +3w=6 4u + 5v+6w=12 6u +9v=4 then show that the roots of the equation ( 11 1) – +-+- (u V W) +[(b—c)2 +(c-a)? +(d – b)2] x+u+v+w= 0 and 20×2 + 10 (a – d)2x – 9= 0 are reciprocals of each other. | 11 |

1245 | ( mathbf{3} times mathbf{8}+mathbf{6} times mathbf{1 1}+mathbf{9} times mathbf{1 4}+dots ) | 11 |

1246 | The sum of three consecutive multiples of 8 is 888, then multiples are (a) 160, 168, 176 (b) 288, 296, 304 (c) 320, 328, 336 (d) 264,272,280. TL – on. L and Unre or in the ratio 5.7 If four | 11 |

1247 | If three positive numbers ( a, b ) and ( c ) are in ( A . P . ) such that ( a b c=8 ), then the minimum possible value of ( b ) is ( A cdot 2 ) B. ( 4^{frac{1}{5}} ) ( mathbf{c} cdot 4^{frac{2}{5}} ) D. | 11 |

1248 | If ( (1+a x)^{n}=1+8 x+24 x^{2}+dots ; ) then ( frac{a-n}{a+n} ) is equal to ( mathbf{A} cdot mathbf{3} ) B. ( frac{-1}{3} ) c. -3 D. | 11 |

1249 | 5. For any odd integer n > 1, nº-(n-1)2+…+(-1)-1 13 = ….. (1996 – 1 M | 11 |

1250 | Find the tenth term of ( G . P .5,25,125 ) | 11 |

1251 | The digits of a 3 -digit number are in ( A P ) and their sum is ( 15 . ) The number obtained by reversing the digits is 594 less than the original number. Find the number. | 11 |

1252 | Calculate sum of eleventh term of the geometric sequence ( 3,6,12,24, dots ) A . 3141 B. 6141 ( c .2141 ) D. 514 | 11 |

1253 | Find the common ratio and ( 10^{t h} ) term of G.P ( frac{-2}{3},-6,-54, dots ) | 11 |

1254 | Find GM of 4,9 ( A cdot 4 ) B. 6 ( c .9 ) D. 7 | 11 |

1255 | Find ( A M ) of 2 and 16 | 11 |

1256 | If the roots of cubic equation ( a x^{3}+ ) ( b x^{2}+c x+d=0 ) are in G.P., then В ( cdot c a^{3}=b d^{3} ) ( mathbf{c} cdot a^{3} b=c^{3} d ) D. ( a b^{3}-c d^{3} ) | 11 |

1257 | ( frac{boldsymbol{b}+boldsymbol{c}-boldsymbol{a}}{boldsymbol{a}}, frac{boldsymbol{c}+boldsymbol{a}-boldsymbol{b}}{boldsymbol{b}}, frac{boldsymbol{a}+boldsymbol{b}-boldsymbol{c}}{boldsymbol{c}} ) are in A.P., then ( frac{1}{a}, frac{1}{b}, frac{1}{c} ) are in A. A.P. в. н.Р c. G. D. A.G.P | 11 |

1258 | Which term of the A.P. ( 121,117,113, ldots . ) is its first negative term? | 11 |

1259 | The natural numbers are written in the form of a triangle as shown below: The sum of numbers in all the n rows. nth | 11 |

1260 | The monthly salaries in rupees of 30 workers in a factory are given below: ( mathbf{5 0 0 0}, mathbf{7 0 0 0}, mathbf{3 0 0 0}, mathbf{4 0 0 0}, mathbf{4 0 0 0}, mathbf{3 0 0 0}, mathbf{3} ) 4000,9000,3000,5000,4000,4000,3 ( 8000,3000,3000,6000,7000,7000, ) Find the mean of monthly salary. | 11 |

1261 | If ( a, b, c ) are in A.P. then ( frac{a-b}{b-c} ) is equal to A ( cdot frac{a}{b} ) B. ( frac{b}{c} ) ( c cdot frac{a}{c} ) D. | 11 |

1262 | Find the common difference of an ( A P ) whose first term is 100 and the sum of whose first six terms is five times the sum of the next six terms. A . 15 B. -10 c. -20 D. 30 | 11 |

1263 | Find the common difference ( d ) for an ( A P ) where ( a_{1}=10 ) and ( a_{20}=466 ) | 11 |

1264 | Find the tenth term of the A.P: ( mathbf{7}, mathbf{1 3}, mathbf{1 9}, mathbf{2 5}, dots ) | 11 |

1265 | If ( 5,0.5,0.05, ldots ) are in GP, then its fourth term is A . 0.05 B. 0.5 c. 0.005 D. 0.0005 | 11 |

1266 | ( 1+frac{1}{4 times 3}+frac{1}{4 times 3^{2}}+frac{1}{4 times 3^{3}} ) is equal to A . 1.120 B. 1.250 c. 1.140 D. 1.160 | 11 |

1267 | Four consecutive terms of a progression are ( 38,30,24,20 . ) The next term of the progression is ( A cdot 18 ) B ( cdot 17 frac{1}{7} ) ( c cdot 16 ) D. none of these | 11 |

1268 | If A.M of two numbers be twice their G.M then the numbers are in the ratio A ( .2: sqrt{3} ) B. ( 2+sqrt{3}: 2-sqrt{3} ) c. ( 2+sqrt{5}: 2-sqrt{5} ) D. 2: | 11 |

1269 | Is the given Progression arithmetic progression?Why ( 2,3,5,7,8,10,15, dots dots dots ) | 11 |

1270 | If ( a, b, c ) are in A.P., then show that the following are also in A.P. ( boldsymbol{b}+boldsymbol{c}, boldsymbol{c}+boldsymbol{a}, boldsymbol{a}+boldsymbol{b} ) | 11 |

1271 | ff ( x^{18}=y^{21}=z^{28}, ) prove that 3 ( mathbf{3} log _{y} boldsymbol{x}, mathbf{3} log _{z} boldsymbol{y}, mathbf{7} log _{boldsymbol{x}} boldsymbol{z} ) form an A.P. | 11 |

1272 | Simplify ( sum_{n=1}^{3} frac{n}{n^{4}+4} ) A . 0.2773 B. 0.3753 c. 0.3353 D. 0.2573 | 11 |

1273 | Illustration 3.99 Prove that in A ABC, tan A + tan B + tan C 23V3, where A, B, C are acute angles. | 11 |

1274 | An AP consists of 23 terms. If the sum of the 3 terms in the middle is 141 and the sum of the last 3 terms is 261 , then the first term is A . 6 B. 5 ( c cdot 4 ) D. 3 E . 2 | 11 |

1275 | The seventh term of a G.P. is 8 times the fourth term and 5th term is ( 48 . ) Find the G.P. | 11 |

1276 | The least value of ( e^{sin ^{-1} x}+e^{cos ^{-1} x} ) is A ( cdot 1+e^{pi / 2} ) B . ( 2 e^{pi / 2} ) ( mathbf{c} cdot e^{pi / 4} ) D ( cdot 2 e^{pi / 4} ) | 11 |

1277 | Four parts of 24 are in A.P. such that the product of extremes is to product of means is 7: 15 then four parts are A. ( frac{3}{2}, frac{9}{2}, frac{15}{2}, frac{21}{2} ) B. ( frac{11}{2}, frac{13}{2}, 9 ) C. ( frac{5}{2}, frac{15}{2}, frac{9}{2}, frac{21}{2} ) D. ( frac{21}{2}, frac{9}{2}, frac{15}{2}, frac{5}{2} ) | 11 |

1278 | Calculate the sum of the series ( 1+3+ ) ( mathbf{5}+mathbf{7}+ldots . mathbf{2} n-mathbf{1} ) | 11 |

1279 | The common ratio of the G.P. ( boldsymbol{a}^{m-boldsymbol{n}}, boldsymbol{a}^{boldsymbol{m}}, boldsymbol{a}^{boldsymbol{m}+boldsymbol{n}} ) is ( mathbf{A} cdot a^{m} ) B. ( a^{-m} ) ( mathbf{c} cdot a^{n} ) ( D cdot a^{-n} ) | 11 |

1280 | The sum of the series ( 1+2.2+3.2^{2}+ ) ( 4.2^{3}+5.2^{4}+.+100.2^{99} ) is ( ? ) B. ( 100.2^{100}+1 ) c. ( $ $ 99.2^{lambda}[100] $ $ ) D. ( 99.2^{100}+1 ) | 11 |

1281 | The ( 4^{t h} ) term of a geometric sequence is ( frac{2}{3} ) and the seventh term is ( frac{16}{81} . ) Find the geometric sequence | 11 |

1282 | If the first term of GP is ( 7, ) its ( n^{t h} ) term is 448 and sum of ( n ) terms is ( 889, ) then find the fifth term of GP | 11 |

1283 | ff ( boldsymbol{x}=mathbf{1}+boldsymbol{a}+boldsymbol{a}^{2}+boldsymbol{a}^{3}+ldots ) and ( boldsymbol{y}= ) ( 1+b+b^{2}+b^{3}+ldots ., ) then show that ( mathbf{1}+boldsymbol{a} boldsymbol{b}+boldsymbol{a}^{2} boldsymbol{b}^{2}+boldsymbol{a}^{3} boldsymbol{b}^{3}+ldots=frac{boldsymbol{x} boldsymbol{y}}{boldsymbol{x}+boldsymbol{y}-mathbf{1}} ) | 11 |

1284 | Sum the following series ( frac{1}{1.4 .7}+frac{1}{4.7 .10}+frac{1}{7.10 .13}+ldots . ) to terms | 11 |

1285 | Let a sequence ( left{a_{n}right} ) be defined by ( boldsymbol{a}_{boldsymbol{n}}=frac{mathbf{1}}{boldsymbol{n}+mathbf{1}}+frac{mathbf{1}}{boldsymbol{n}+mathbf{2}}+frac{mathbf{1}}{boldsymbol{n}+mathbf{3}}+dots dots+ ) ( frac{1}{3 n}, ) then This question has multiple correct options A ( a_{2}=frac{7}{12} ) B. ( a_{2}=frac{19}{20} ) ( ^{mathbf{C}} cdot a_{n+1}-a_{n}=frac{9 n+5}{(3 n+1)(3 n+2)(3 n+3)} ) D. ( a_{n+1}-a_{n}=frac{-2}{3(n+1)} ) | 11 |

1286 | Find the sum to 20 terms in each of the geometric progressions in ( mathbf{0 . 1 5}, mathbf{0 . 0 1 5}, mathbf{0 . 0 0 1 5}, ldots ) | 11 |

1287 | ( a, b, c, d, e ) are in ( A . P . ) Prove the following results ( boldsymbol{a}-boldsymbol{b}+boldsymbol{c}-boldsymbol{d}=mathbf{0} ) | 11 |

1288 | Which one of the following is a geometric progression? A. 3,5,9,11,15 в. 4,-4,4,-4,4 c. 12,24,36,48 D. 6,12,24,36 | 11 |

1289 | Find the ( 20^{t h} ) term of an A.P. whose ( 5^{t h} ) term is 15 and the sum of its ( 3^{r d} ) and ( 8^{t h} ) terms is 34 | 11 |

1290 | The arithmetic mean between ( 2+sqrt{(2)} ) and ( 2-sqrt{(2)} ) is A .2 B. ( sqrt{(2)} ) c. 0 D. | 11 |

1291 | ( ln operatorname{an} A cdot P-10,-6,-2,2, dots 20 t h ) term is A . 66 B. -66 ( c cdot 77 ) D. None of these | 11 |

1292 | Which term of the sequence 3,6,12 is ( 1536 ? ) | 11 |

1293 | f ( p ) is a prime, show that the sum of the ( (p-1)^{t h} ) powers of any ( p ) numbers in arithmetical progression, wherein the common difference is not divisible by ( boldsymbol{p} ) is less by 1 than a multiple of ( p ) | 11 |

1294 | If ( p, q, r, s ) are in ( A . P ) ad if ( (x)= ) [ begin{array}{rl} boldsymbol{p}+sin boldsymbol{x} & boldsymbol{q}+sin boldsymbol{x}-boldsymbol{r}+sin boldsymbol{x} \ | boldsymbol{q}+sin boldsymbol{x} & boldsymbol{r}+sin boldsymbol{x} quad-mathbf{1}+sin boldsymbol{x} \ boldsymbol{r}+sin boldsymbol{x} & boldsymbol{s}+sin boldsymbol{x} quad boldsymbol{s}-boldsymbol{q}+boldsymbol{s} boldsymbol{i} boldsymbol{n} boldsymbol{x} end{array} ] such that ( int_{0}^{2} f(x) d x=-4 ) then the common difference of the ( A . P . ) cn A. -1 B. c. 1 D. none | 11 |

1295 | After dividing each term of the arithmetic progression with a fixed number, the common difference is A. Added by the same fixed number. B. Divided by the same fixed number. c. Multiplied by another fixed number. D. Multiplied by the same fixed number | 11 |

1296 | Illustration 3.39 Prove that SC_ 1 cos 0 + cos(2r +1)0) _ sin ne 2 sin 0.cos cos: (n+1)0” (where r= ne M | 11 |

1297 | ff ( x, 2 x+2,3 x+3, ldots ldots ) are in G.P., then the fourth term is A . -13. B. 25.1 ( c .-12.6 ) D. 17.6 | 11 |

1298 | Answer the following The first term of an AP is -7 and the common difference ( 5, ) Find its ( 18^{t h} ) term | 11 |

1299 | Find the common difference ( d ) and write three more terms.2, ( frac{5}{2}, 3, frac{7}{2} ) | 11 |

1300 | If the geometric progression ( 162,54,18, dots ) and ( frac{2}{81}, frac{2}{27}, frac{2}{9}, dots . . ) here their nth term equal. Find the value of ( n ) | 11 |

1301 | If the ( 10^{t h} ) term of an A.P. is 52 and the ( 17^{t h} ) term is 20 more than the ( 13^{t h} ) term, find the A.P. | 11 |

1302 | The first term and the common difference of the arithmetic progression ( mathbf{3}, mathbf{1 0}, mathbf{1 7}, mathbf{2 4}, dots ) is? A. 7 and 3 B. 3 and 7 c. 3 and 10 D. Not defined | 11 |

1303 | How many leap years are there in set of any consecutive 400 years? | 11 |

1304 | The sum of first 5 odd numbers is called A. term B. constant c. series D. sequence | 11 |

1305 | harmonic mean of two numbers is 4. Their arithmetic an A and the geometric mean G satisfy the relation. 2A+8=27 Find the two numbers. (1979) Arithmetic | 11 |

1306 | Find the common difference of the A.P. ( boldsymbol{y}-mathbf{7}, boldsymbol{y}-mathbf{2}, boldsymbol{y}+mathbf{3} ) | 11 |

1307 | ( ln operatorname{an} A cdot P ) if ( a_{1}=2, a_{3}=26 ) Find ( a_{2} ) | 11 |

1308 | The reciprocals of all the terms of a geometric progression form a progression. ( A cdot A P ) в. нР c. ศ D. AGP | 11 |

1309 | The minimum value of ( mathbf{a}^{2} sec ^{2} boldsymbol{theta}+ ) ( mathbf{b}^{2} cos mathbf{e c}^{2} boldsymbol{theta} ) is ( A cdot a^{2}-b^{2} ) B. ( 2left(a^{2}+b^{2}right) ) ( c cdot(a-b)^{2} ) D. ( (a+b)^{2} ) | 11 |

1310 | In an A.P., of which a is the first term, if the sum of the first p terms is zero, show that the sum of the next q terms is ( -frac{a(p+q) q}{p-1} ) | 11 |

1311 | Find nth term of the ( boldsymbol{G} cdot boldsymbol{P} cdot sqrt{boldsymbol{3}}, frac{mathbf{1}}{sqrt{mathbf{3}}}, frac{1}{mathbf{3} sqrt{mathbf{3}}}, ldots ) | 11 |

1312 | The ( 2^{n d}, 31^{s t}, ) and last terms of an A.P. ( operatorname{are} 7 frac{3}{4}, frac{1}{2} ) and ( -6 frac{1}{2} ) respectively; find the first term and the number of terms. | 11 |

1313 | Find the Odd one among : 49, 121, 169, 225,289,361 A . 225 B. 121 ( c cdot 49 ) D. 36 | 11 |

1314 | The sum of all integers between 81 and 719 which are divisible by 5 is ( mathbf{A} cdot 51800 ) B. 50800 c. 52800 D. None of these | 11 |

1315 | The nth term of the A.P. ( 3,7,11,15, ldots . ) is given by – A ( cdot T_{n}=4 n-1 ) в. ( T_{n}=3 n+4 ) c. ( T_{n}=3 n-4 ) D. ( T_{n}=2 n+1 ) | 11 |

1316 | Prove that the sequence, whose general ( operatorname{term} ) is ( a_{n}=2.3^{n}, ) is a geometric progression and find the stint of the first eight terms. | 11 |

1317 | If a A.P. ( & ) an H.P. have the same first term, the same last term & the same number of terms; then the product of the ( r^{t h} ) term from the beginning in one series ( & ) the ( r^{t h} ) term from the end in the other is independent of ( r ) If true enter 1 else enter 0 | 11 |

1318 | (C) 100 12 34. Let a,,a,a 3…., 249 be in A.P. such that 24k+1 = 416 k=0 and ag +843 = 66. If a1 + a2 +…+ a1 = 140m , then mis equal to : [JEE M 2018 (a) 68 (b) 34 © 33 d) 66 | 11 |

1319 | Find the sum of the following series. ( 1+(1+a)+r+left(1+a+a^{2}right) r^{2}+dots ) to ( infty(|boldsymbol{r}|<mathbf{1}) ) | 11 |

1320 | For the following arithmetic progressions write the first term ( a ) and the common difference ( d ) : ( mathbf{0 . 3}, mathbf{0 . 5 5}, mathbf{0 . 8 0}, mathbf{1 . 0 5}, ldots ) | 11 |

1321 | ff ( x, y, z ) are positive, then prove that ( (x+y+z)left(frac{1}{x}+frac{1}{y}+frac{1}{z}right) geq 9 ) | 11 |

1322 | (a) 142 (6) 192 (c) ? Is the A.M. of two distinct real numbers I and nl, n>1) and G1, G2 and G, are three geometric means between land n, then G* +267 +G; equals : JEEM 2015] (a) 4 lmn2 (b) 412m²n2 (c) 41mn (d) 4 Im?n | 11 |

1323 | Sum of infinity the following series: ( mathbf{3}+mathbf{5} cdot frac{mathbf{1}}{mathbf{4}}+mathbf{7} cdot frac{mathbf{1}}{mathbf{4}^{2}}+dots dots ) | 11 |

1324 | Find the Arithmetic mean between 24 and 36 A . 26 B. 28 ( c .30 ) D. 32 | 11 |

1325 | Sum of the first ( p, q ) and ( r ) terms of an A.P. are ( a, b ) and ( c, ) respectively. Prove ( operatorname{that} frac{boldsymbol{a}}{boldsymbol{p}}(boldsymbol{q}-boldsymbol{r})+frac{boldsymbol{b}}{boldsymbol{q}}(boldsymbol{r}-boldsymbol{p})+ ) ( frac{c}{r}(p-q)=0 ) | 11 |

1326 | ( operatorname{Let}left(1+x^{2}right)^{2}(1+x)^{n}=sum_{k=0}^{n+4} a_{k} x^{k} ) ( a_{1}, a_{2} ) and ( a_{3} ) are in arithmetic progression, find ( n ) | 11 |

1327 | Find the eleventh term of the A.P. ( 2,7,12, dots ) | 11 |

1328 | If the sum of the roots of the equation ( a x^{2}+b x+c=0 ) is equal to sum of the squares of their reciprocals, then ( b c^{2}, c a^{2}, a b^{2} ) are in A . ( A . P ) в. ( G . P ) c. ( H . P ) D. A.G.P | 11 |

1329 | Sum of infinite number of terms of GP is 20 and sum of their square is 100. The common ratio of GP is [2002] (a) 5 (6) 315 (c) 8/5 (d) 1/5 | 11 |

1330 | The fourth term of a G.P. is 27 and the 7th term is ( 729, ) find the G.P. | 11 |

1331 | If 6 arithmetic means are inserted between 1 and ( frac{9}{2}, ) then find the ( 4^{t h} ) arithmetic mean. | 11 |

1332 | The sum of first ( n ) terms of an AP is given by ( S_{n}=left(1+T_{n}right)(n+2) ) Then ( T_{2} ) is A ( cdot-frac{11}{6} ) B. ( -frac{-5}{3} ) ( c cdot-frac{5}{3} ) D. 2 | 11 |

1333 | What is the sum of 12 odd numbers ( mathbf{1}, mathbf{3}, mathbf{5}, mathbf{7}, mathbf{9} ldots . . ? ) A . 12 B. 144 ( c .141 ) D. 124 | 11 |

1334 | Find the number of terms in an arithmetic progression with the first term 2 and the last term being ( 62, ) given that common difference is 2 A . 31 B. 40 ( c cdot 22 ) D. 27 | 11 |

1335 | If ( a_{r}>0, r epsilon N ) and ( a_{1}, a_{2}, a_{3}, dots, a_{2 n} ) are in AP then ( frac{boldsymbol{a}_{1}+boldsymbol{a}_{2 n}}{sqrt{boldsymbol{a}_{1}}+sqrt{boldsymbol{a}_{2}}}+frac{boldsymbol{a}_{2}+boldsymbol{a}_{2 n-1}}{sqrt{boldsymbol{a}_{2}}+sqrt{boldsymbol{a}_{3}}}+ ) ( frac{boldsymbol{a}_{boldsymbol{3}}+boldsymbol{a}_{boldsymbol{2} boldsymbol{n}-boldsymbol{2}}}{sqrt{boldsymbol{a}_{boldsymbol{3}}}+sqrt{boldsymbol{a}_{boldsymbol{4}}}}+ldots+frac{boldsymbol{a}_{boldsymbol{n}}+boldsymbol{a}_{boldsymbol{n}+1}}{sqrt{boldsymbol{a}_{boldsymbol{n}}}+sqrt{boldsymbol{a}_{boldsymbol{n}+mathbf{1}}}} ) is equal to A . n- B. ( frac{nleft(a_{1}+a_{2 n}right)}{sqrt{a_{1}}+sqrt{a_{n+1}}} ) c. ( frac{n-1}{sqrt{a_{1}}+sqrt{a_{n+1}}} ) D. none of these | 11 |

1336 | For ( a n A, P, frac{1}{3}, frac{4}{3}, frac{7}{3}, frac{10}{3} dots dots dots, ) Find ( T_{18} ) For ( a n A, P, 3,9,15,21 dots dots dots, ) Find ( S_{10^{circ}} ) | 11 |

1337 | Find ( A M ) of 5 and 23 | 11 |

1338 | In a sequence, if ( S_{n} ) is the sum of the first ( n ) terms and ( S_{n-1} ) is the sum of the first ( (n-1) ) terms, then the ( n^{t h} ) term is A. ( S_{n-2} ) в. ( S_{n}-S_{n-1} ) ( mathbf{c} cdot S_{n+1}-S_{n} ) D. ( S_{n+1}-S_{n-1} ) | 11 |

1339 | If the arithmetic mean between ( a ) and ( b ) is twice as great as the geometric mean, show that ( a: b=2+sqrt{3}: 2- ) ( sqrt{3} ) | 11 |

1340 | Insert 20 AM between 2 and 86 .Then find first mean. | 11 |

1341 | Find the ( 9^{t h} ) term of an A.P ( mathbf{2}, mathbf{5}, mathbf{8}, mathbf{1 1}, dots dots ) | 11 |

1342 | If the sum of ( n ) terms of an A.P.is ( n P+ ) ( frac{1}{2} n(n-1) Q, ) where ( P ) and ( Q ) are constants, find the common difference. | 11 |

1343 | Prove ( left(1+frac{1}{1}right)left(1+frac{1}{2}right)left(1+frac{1}{3}right) dotsleft(1+frac{1}{n}right) ) ( (n+1) ) | 11 |

1344 | When a number ( x ) is subtracted from each of the numbers ( 8,16, ) and ( 40, ) the resulting three numbers form a geometric progression. Find the value of ( boldsymbol{x} ) A . 3 B. 4 c. 6 D. 12 E . 18 | 11 |

1345 | ff ( x_{1}, x_{2}, dots, x_{n} ) are ( ^{prime} n^{prime} ) positive real numbers; then ( A . M geq G . M geq H . M ) ( frac{x_{1}+x_{2}+x_{3}+ldots+x_{n}}{n} geqleft(x_{1} x_{2} x_{3} ldots x_{n}right)^{n} geq ) ( frac{n}{frac{1}{x_{1}}+frac{1}{x_{2}}+frac{1}{x_{3}}+ldots+frac{1}{x_{n}}} ) equality occurs when numbers are same using this concept.Here equality occurs numbers are of same.Using this concept answer the following f ( a>0, b>0, c>0 ) and the minimum value of ( aleft(b^{2}+c^{2}right)+bleft(c^{2}+a^{2}right)+ ) ( cleft(a^{2}+b^{2}right) ) is ( lambda a b c, ) then ( lambda ) is ( mathbf{A} cdot mathbf{1} ) B. 2 ( c .3 ) D. | 11 |

1346 | The sum of first ( n ) terms of an AP is given by ( S_{n}=5 n^{2}+3 n, ) then find its nth term. | 11 |

1347 | Find GM between 4 and 36 | 11 |

1348 | ( operatorname{Let} a_{1}, a_{2}, dots dots a_{50} ) are non constant terms of an A.P. and sum of n terms is given by ( boldsymbol{S}_{boldsymbol{n}}=mathbf{5 0 n}+(boldsymbol{n})(boldsymbol{n}-boldsymbol{7}) frac{boldsymbol{A}}{mathbf{2}} ) then ordered pair ( left(d, a_{50}right) ) is? (where dis the common difference A. ( (A, 45 A) ) B. ( (46 A+50) ) c. ( (2 A, 46 A) ) D. ( (2 A, 50+49 A) ) | 11 |

1349 | 10. The sum of the first n terms of the series 12 +2.22 +32 +2.42 +52 +2.62 + n(n+1) when n is even. When n is odd the sum is is 2 [20 @[109,7″) | e n(n + 1) () r? (n +1) (a) 3n(n + 1) 2 n(n + 3n(n+1) | 11 |

1350 | If the numbers ( a, b, c, d, e ) form an A.P. then the value of ( a-4 b+6 c-4 d+e ) is? ( A cdot 1 ) B . 2 ( c cdot 0 ) D. None of these | 11 |

1351 | ( lim _{n rightarrow infty} frac{1^{P}+2^{P}+3^{P}+ldots .+n^{P}}{n^{P+1}} ) equals- | 11 |

1352 | 14. mn squares of euqal size are arranged to from a rectangle of dimension m by n, where m and n are natural numbers. Two squares will be called ‘neighbours’ if they have exactly one common side. A natural number is written in each square such that the number written in any square is the arithmetic mean of the numbers written in its neighbouring squares. Show that this is possible only if all the numbers used are equal. (1982-5 Marks) | 11 |

1353 | ( ln a G P ) the ( 3^{r d} ) term is 24 and the ( 6^{t h} ) term is ( 192 . ) Find the ( 10^{t h} ) term. | 11 |

1354 | suitable common differences given in column B. Column A Column B A. ( -2,2,6,10, dots ) 1. ( frac{2}{3} ) B. ( a=18, n=10, a_{n}=0 ) 2. -5 ( mathbf{C} cdot boldsymbol{a}=mathbf{0}, boldsymbol{a}_{mathbf{1 0}}=mathbf{6} ) 3. 4 D. ( boldsymbol{a}_{mathbf{2}}=mathbf{1 3}, boldsymbol{a}_{boldsymbol{4}}=mathbf{3} ) 4. -4 ( 5 .-2 ) 6. ( frac{1}{2} ) 7. 5 A. A-3, B-5, C-1, D-2 B. A-4, B-1, C-3, D-2 C. A-2, B-7, C-5, D-6 D. ( A-7, B-4, C-2, D-2 ) | 11 |

1355 | Split 69 late three pats such that they are in ( A . P . ) and the product of two smaller parts is 483 A . 21 B. 23 c. 25 D. Allofabove | 11 |

1356 | Sum of ( frac{1}{2.7}+frac{1}{7.12}+frac{1}{12.17}+ ) ( frac{1}{17.22}+ldots ) to ( n ) terms | 11 |

1357 | If ( frac{a-b}{b-c}=frac{a}{b}, ) then ( a, b, c ) are in ( A cdot G P ) в. нР ( c cdot A P ) D. sp | 11 |

1358 | If ( boldsymbol{x}, boldsymbol{y}, boldsymbol{z} ) are in A.P. and A.M. of ( boldsymbol{x} ) and ( boldsymbol{y} ) is a and that to ( y ) and ( z ) is ( b ), then ( A . M ). of ( a ) and ( b ) is. ( A cdot x ) B. ( y ) ( c cdot z ) D. ( frac{1}{2(x+y)} ) | 11 |

1359 | The arithmetic mean between two distinct positive numbers is twice the geometric mean between them. Find the ratio of greater to smaller. | 11 |

1360 | If the ( m^{t h} ) term of an ( A . P . ) is ( frac{1}{n} ) and ( n^{t h} ) term is ( frac{1}{m}, ) then the sum of first ( m n ) terms is A ( . m n+1 ) в. ( frac{m n+1}{2} ) c. ( frac{m n-1}{2} ) D. ( frac{m n-1}{3} ) | 11 |

1361 | The A.M. of the first ten odd numbers is A . 10 B. 100 c. 1000 D. | 11 |

1362 | Find the ( 6^{t h} ) term from the end of the A.P: ( mathbf{1 7}, mathbf{1 4}, mathbf{1 1}, dots dots-mathbf{4 0} ) | 11 |

1363 | A man saved RS. 200 in each of the first three months of his service. In each of the subsequent months his saving increases by Rs. 40 more than the saving of immediately previous month. His total saving from the start of service will be Rs. 11040 after. A. 18 months B. 19 months c. 20 months D. 21 months | 11 |

1364 | ( mathbf{1}+mathbf{4}+mathbf{1 0}+mathbf{1 9}+ldots frac{mathbf{3} boldsymbol{n}^{2}-mathbf{3} boldsymbol{n}+mathbf{2}}{mathbf{2}}= ) A ( cdot frac{n^{2}left(n^{2}+1right)}{2} ) B ( cdot frac{nleft(n^{2}+1right)}{2} ) c. ( frac{n^{2}(n+1)}{2} ) D. ( left{frac{n(n+1)}{2}right}^{2} ) | 11 |

1365 | If the ( p^{t h}, q^{t h} ) and ( r^{t h} ) terms of an A.P. are ( P, Q, R ) respectively, then ( P(q-r)+ ) ( Q(r-p)+R(p-q) ) equals A. B. c. par D. p ( + ) ar | 11 |

1366 | Which term of the A.P. ( 3,15,27,39, dots ) will be 132 more than its ( 54^{t h} ) term? | 11 |

1367 | Find ( A M ) of first 250 natural numbers A. 115 в. 225 ( c cdot 125 ) D. 125.5 | 11 |

1368 | The value of ( 2^{n}{1.3 .5 ldots ldots .(2 n- ) ( mathbf{3})(mathbf{2 n – 1})} ) is A ( cdot frac{(2 n) !}{n !} ) B. ( frac{(2 n) !}{2^{n}} ) c. ( frac{n !}{(2 n !) !} ) D. None of these | 11 |

1369 | Directions: In following quesiton, choose the missing term out of the given alternatives Reference: ( A B C D E F G H I J K L M ) N O P Q R ST U V W X Y Z ( Z, X, V, T, R, dots dots ) A. ( O, K ) в. ( N, M ) c. ( K, S ) D. ( M, N ) E . ( P, N ) | 11 |

1370 | The arithmetic mean of first five natural number is A . 2 B. 3 ( c cdot 4 ) ( D ) | 11 |

1371 | If the distinct points on the curve ( y= ) ( 2 x^{4}+7 x^{3}+3 x-5 ) are collinear, then find the arithmatic mean of ( x- ) coordinates of the aforesaid points | 11 |

1372 | The sum of first ( n ) terms of an G.P. is A ( cdot s_{n}=frac{a_{1}left(1-r^{n}right)}{1-r} ) B. ( s_{n}=frac{a_{1}left(1+r^{n}right)}{1-r} ) ( ^{mathbf{c}} cdot_{S_{n}}=frac{a_{1}left(1-r^{n}right)}{1+r} ) D. ( _{S_{n}}=frac{a_{1}left(1-r^{n}right)}{r-1} ) | 11 |

1373 | Find the first term ( a_{1} ) and the common difference ( d ) of the arithmetic progression in which ( boldsymbol{a}_{mathbf{2}}+boldsymbol{a}_{mathbf{5}}-boldsymbol{a}_{mathbf{3}}= ) ( mathbf{1 0}, boldsymbol{a}_{mathbf{2}}+boldsymbol{a}_{mathbf{9}}=mathbf{1 7} ) | 11 |

1374 | If ( a+3 b+2 c=12 ), then the maximum value of ( 6 b c(1+a)+a(3 b+2 c) ) is A. 112 в. 115 ( c cdot 102 ) D. 92 | 11 |

1375 | Find the G.M of 6 and 24 | 11 |

1376 | Find the geometric mean of ( sqrt{82}-1 ) and ( sqrt{82}+1 ) ( A cdot S ) B. ( c cdot 81 ) D. 27 | 11 |

1377 | If ( S=frac{1}{1.5}+frac{1}{5.9}+frac{1}{9.13}+ldots . ., ) then ( ^{mathbf{A}} cdot T_{n}=frac{1}{4}left(frac{1}{4 n-3}-frac{1}{4 n+1}right) ) B. ( s_{n}=frac{1}{4}left[1-frac{1}{4 n+1}right] ) c. ( S_{infty}=1 ) D. None | 11 |

1378 | The sum of the series ( 1+frac{1}{6}+frac{1}{18}+ ) ( frac{7}{324}+ldots . infty, ) is | 11 |

1379 | Give the next decimal numbers the sequence: ( (a) 1.32,1.42,1.52, dots dots . . ) (b) ( 1.14,1.25,1.36, dots dots . . . ) | 11 |

1380 | Identify which of the following list of numbers is an arithmetic progression? A ( .1,1,2,3,5, ldots ) в. ( 2,3,5,7,11, ldots ) c. ( 10,100,1000, ldots ) D. ( 12,18,24,30, ldots ) | 11 |

1381 | RS) 22. Let V be the volume of the parallelopiped formed by th vectors a = a[i+az j +azk, b = bli +b29 +b3k. Č=+c2j+czł . If a,, b, c,, where r= 1, 2, 3, are non- negative real numbers and + br +Cr) = 3L, show w r=1 that V SĽ. (2002 – 5 Marks) | 11 |

1382 | The general term of a sequence is given by ( a_{n}=-4 n+15 . ) Is the sequence an A.P? If so, find its 15 th term and the common difference. | 11 |

1383 | Is 51 a term of the ( A P, 5,8,11,14, ldots ldots . . ? ) A. Yes B. No c. Ambiguous D. Data insufficient | 11 |

1384 | Find the ( n^{t h} ) term and the ( 7^{t h} ) term of ( mathbf{3},-mathbf{6},+mathbf{1 2},-mathbf{2 4}, ldots ) A . 191 в. 193 ( c cdot 192 ) D. 194 | 11 |

1385 | If ( a, b ) and ( c ) arc in arithmetic progression then ( frac{b-a}{c-b} ) is equal to ( A cdot frac{b}{a} ) B. ( c cdot 1 ) D. ( 2 a ) | 11 |

1386 | Find the 9 th term and the general term of the progression: ( frac{1}{4},-frac{1}{2}, 1,-2, dots dots ) | 11 |

1387 | If S1,S2,S3, ……………, Sn are the sums of infinite geometric ories whose first terms are 1, 2, 3, ……………, n and whose 1 1 1 common ratios are 2’3’4…… – respectively, ntl then find the values of Sı? +S2? +S3? + ………… + S3n-1 (1991- A Model | 11 |

1388 | Find the sum the infinite G.P.: ( frac{2}{3}-frac{4}{9}+frac{8}{27}-frac{16}{21}+dots dots ) A ( cdot frac{2}{5} ) B. ( frac{3}{5} ) c. ( frac{19}{27} ) D. 8 | 11 |

1389 | 14. The fourth power of the common difference of an arithmatic progression with integer entries is added to the product of any four consecutive terms of it. Prove that the resulting sum is the square of an integer. (2000 – 4 Marks) | 11 |

1390 | I: The real number x when added to its inverse gives the minimum positive value of the sum at ( x=1 ) Il: If product of the two positive numbers is 400 , then the minimum value of their sum is 20 which of the above statements are true A . only lis true B. only II is true c. both I and II are true D. niether I nor II are true | 11 |

1391 | If an ( A . P . ) has 21 terms, and the sum of ( 10^{t h}, 11^{t h} & 12^{t h} ) terms is 129 and sum of last three terms is 237 . Find first term. | 11 |

1392 | ( frac{1}{n} sum_{i=1}^{n} f^{-1}left(x_{i}right)=fleft(frac{1}{n} sum_{i=1}^{n} x_{i}right) ) | 11 |

1393 | If the lengths of the sides of a right angled triangle ( A B C ) right angled at ( C ) ( operatorname{are} operatorname{in} A . P ., ) find ( 5(sin A+sin B) ) | 11 |

1394 | Verify the ( boldsymbol{A P}: sqrt{mathbf{3}}, mathbf{2} sqrt{mathbf{3}}, mathbf{3} sqrt{mathbf{3}}, ldots ) and find the product of the next two terms: | 11 |

1395 | The sums of ( n ) terms of two arithmetic progressions are in the ratio ( 5 n+4 ) ( 9 n+6 . ) Find the ratio of their 18 th terms. | 11 |

1396 | Let ( t_{r}=frac{r}{1+r^{2}+r^{4}} ) then, ( lim _{n rightarrow infty} sum_{r=1}^{n} t_{r} ) equals ( A cdot frac{1}{4} ) B. 1 ( c cdot frac{1}{2} ) D. None of these | 11 |

1397 | Find the value of the sum ( sum_{r=1}^{n} ) ( sum_{s=1}^{n} delta_{r s} 2^{r} 3^{s} ) where ( delta_{r s} ) is zero if ( r neq s & ) ( delta_{r s} ) is one if ( r=s ) A ( cdot frac{6left(6^{n}-1right)}{5} ) B. ( frac{6left(6^{n}+1right)}{5} ) c. ( frac{5left(6^{n}+1right)}{6} ) D. ( frac{nleft(6^{n}-1right)}{6} ) | 11 |

1398 | Choose the correct answer from the alternatives given : Given ( 1+2+34++10=55 ) then the ( operatorname{sum} 6+12+18+24++60 ) is equal to A . 300 B. 655 ( c .330 ) D. 455 | 11 |

1399 | Find the sum of the following ( boldsymbol{A P} ) ( frac{1}{15}, frac{1}{12}, frac{1}{10}, dots ., ) to 11 terms | 11 |

1400 | For the A.P ( 45,41,37, ldots ) find ( t_{10} ) and ( boldsymbol{t}_{boldsymbol{n}+mathbf{1}} ) | 11 |

1401 | If ( s ) represents the sum of ( n ) terms of G.P whose first term and common ratio are ( a ) and ( r ) respectively, then ( s_{1}+s_{2}+ ) ( boldsymbol{s}_{3}+ldots+boldsymbol{s}_{boldsymbol{n}} ) | 11 |

1402 | If ( a^{2}+b^{2}, a b+b c ) and ( b^{2}+c^{2} ) are in G.P. ( a, b, c ) are in : A . A.P. B. G.P. c. н.P. D. cannot be determined | 11 |

1403 | If ( frac{p+q}{p-q}, x quad ) and ( quad p^{2}-q^{2} ) are in continued proportion the find ( x ) | 11 |

1404 | Find: the ninth term of the G.P. ( mathbf{1}, mathbf{4}, mathbf{1 6}, mathbf{6 4}, dots ) | 11 |

1405 | Assertion Statement-1 If ( a_{1}, a_{2}, a_{3}, dots dots dots, a_{24} ) are ( ln ) A. P. such that ( a_{1}+a_{5}+a_{10}+a_{15}+ ) ( boldsymbol{a}_{20}+boldsymbol{a}_{24}=mathbf{2 2 5} ) then ( boldsymbol{a}_{mathbf{1}}+boldsymbol{a}_{mathbf{2}}+boldsymbol{a}_{3}+ ) ( ldots .+a_{23}+a_{24}=900 ) because Reason Statement-2 In any A.P. sum of the terms equidistant from begining and end is constant and is equal to the sum of the first and the last term, A. Statement-1 is true, Statement-2 Is true and statement-2 is correct explanation for Statement- B. Statement-1 is true, Statement-2 Is true and Statement-2 is NOT correct explanation for Statement c. statement-1 is true, statement-2 is false D. statement-1 is false, Statement-2 Is true | 11 |

1406 | How many terms of the ( boldsymbol{A} . boldsymbol{P} ) ( -6,-frac{11}{2},-5, dots ) are needed to give the sum ( -25 ? ) | 11 |

1407 | Find the value of ( frac{1}{1+} frac{2}{2+} frac{3}{3+} cdots ) | 11 |

1408 | Identify the geometric progression. A ( .1,3,5,7,9, dots ) в. 2,4,6,8,10 ( mathbf{c} cdot 5,10,15,25,35 ) D. ( 1,3,9,27,81 dots ) | 11 |

1409 | The ( 11^{t h} ) term and the ( 21^{s t} ) term of an A.P. are 16 and 29 respectively, then find the ( 34^{t h} ) term | 11 |

1410 | Sum of the series ( 1+(1+2)+ ) ( (1+2+3)+(1+2+3+4)+ldots ) to ( n ) terms be ( frac{1}{m} n(n+1)left[frac{2 n+1}{k}+1right] ) Find the value of ( mathrm{k}+mathrm{m} ) | 11 |

1411 | The sum of first 20 odd natural number is : A . 100 в. 210 c. 400 D. 420 | 11 |

1412 | Find the A.P whose ( 3^{r d} ) term is 16 and the ( 7^{t h} ) term exceeds its ( 5^{t h} ) term by 12 | 11 |

1413 | If ( 1, log _{3}left(3^{x}-2right), 2 log _{9}left(3^{x}-8 / 3right) ) are in A.P., then the value of ( x ) can be- ( mathbf{A} cdot log left(frac{4}{3}right) ) B. ( frac{log 4}{log 3} 4 ) c. ( mathbf{D} cdot log _{3} 4 ) | 11 |

1414 | Sum of the first n terms of the series 1 3 7 15 -+-+-+- +………….. is equal to (1988 – 2 Marks) 2 4 8 16 (a 2″ – -1 (b) 1-2-n (c) nt 2-n-1 (d) 2″ +1. | 11 |

1415 | The ( 17^{t h} ) term of an ( A . P ) succeed ( 10^{t h} ) term by ( 7 . ) Find common difference ( d ) ? | 11 |

1416 | Find the sum of all natural numbers from 1 to 200 which are divisible by 4 | 11 |

1417 | ff ( l=20, d=-1, n=17, ) then the first term is : A . 30 B. 32 ( c .34 ) D. 36 6 | 11 |

1418 | Sum of first 14 terms of an ( A P ) is 1505 and it first term is ( 10 . ) Find its ( 25^{t h} ) term | 11 |

1419 | The common ratio for the term ( boldsymbol{a}_{n}= ) ( 2 timesleft(frac{1}{4}right)^{n-1} ) is A ( cdot frac{1}{2} ) в. ( frac{1}{3} ) ( c cdot frac{1}{4} ) D. | 11 |

1420 | The AM of multiple of 5 from numbers 1 to 500 is ( mathbf{A} cdot 250 ) в. ( frac{500}{2} ) c. ( frac{505}{2} ) D. 252.5 | 11 |

1421 | Find the ( n^{t h} ) term of ( 6,11,16,21, dots dots ) | 11 |

1422 | Find the next term of the sequence: ( 2,5,8,11, dots dots dots ) A . 14 B . 15 c. 16 D. 18 | 11 |

1423 | If 8 G.M.’s are inserted between 2 and 3 then the product of the 8 G.M.’s is ( mathbf{A} cdot mathbf{6} ) B. 36 c. 216 D. 1296 | 11 |

1424 | Suppose a population ( A ) has 100 observations ( 101,102 ldots 200 ) and other population ( B ) has 100 observations ( 151,152 ldots 250 ) Find the difference in their means A . 49 B . 50 c. 51 D. 52 | 11 |

1425 | It is known that ( sum_{r=1}^{infty} frac{1}{(2 r-1)^{2}}=frac{pi^{2}}{8}, ) then ( sum_{r=1}^{infty} frac{1}{r^{2}} ) is equal to A ( cdot frac{pi^{2}}{24} ) B. ( frac{pi^{2}}{3} ) c. ( frac{x^{2}}{6} ) D. none of these | 11 |

1426 | Find the ( 12^{t h} ) term of the A.P.: ( mathbf{9}, mathbf{1 3}, mathbf{1 7}, mathbf{2 1}, dots . ) | 11 |

1427 | ( mathbf{A} ) is the sum of some set of terms of a sequence. A. term B. constant c. series D. sequence | 11 |

1428 | Find out which of the following sequence are arithmetic progression. For those which are arithmetic progression, find out the common difference. ( frac{1}{2}, frac{1}{4}, frac{1}{6}, frac{1}{8}, dots dots ) | 11 |

1429 | The interior angles of a polygon are in arithmetic progression. The smallest angle is 120°, and the common difference is 5º, Find the number of sides of the polygon. (1980 | 11 |

1430 | The A.M. of first n even natural number is A ( . n(n+1) ) в. ( frac{n+1}{2} ) ( c cdot frac{n}{2} ) D. ( n+1 ) | 11 |

1431 | If the sum of a certain number of terms of the ( A P 25,22,19, ldots . ) is ( 116 . ) Find the last term. | 11 |

1432 | If the third term of a G.P. is 12 and its sixth term is ( 96, ) find the sum of 9 terms. | 11 |

1433 | The nth term of a sequence is ( 3 n-2 . ) Is the sequence an A.P.? If so, find its 10 th term | 11 |

1434 | If ( (1-p)left(1+3 x+9 x^{2}+27 x^{3}+81 x^{4}+right. ) ( 1-p^{6}, p neq 1 ) then the value of ( frac{p}{x} ) is A ( cdot frac{1}{3} ) B. 3 ( c cdot frac{1}{2} ) D. 2 | 11 |

1435 | Two arithmetic progressions have the same common difference. The difference between their 100 th terms is 100, what is the difference between their 1000 th terms? | 11 |

1436 | 20. [2009 The sum to infinite term of the series 2 6 10 14 . 1+ – +- +- … is +- + 13 32 33 34 (a) 3 (6) 4 (c) 6 . (d) 2 | 11 |

1437 | If the roots of ( a x^{3}+b x^{2}+c x+d=0 ) are in A.P., then the roots of ( a(x+ ) ( k)^{3}+b(x+k)^{2}+c(x+k)+d=0 ) are in A. A.P в. G. c. н. D. A.G.P. | 11 |

1438 | If ( S_{1}, S_{2} ) and ( S_{3} ) are the sum of first ( boldsymbol{n}, mathbf{2 n} ) and ( mathbf{3 n} ) terms of a geometric series respectively, then prove that ( boldsymbol{S}_{1}left(boldsymbol{S}_{3}-boldsymbol{S}_{2}right)=left(boldsymbol{S}_{2}-boldsymbol{S}_{1}right)^{2} ) | 11 |

1439 | Find the sum of the series ( 3+7+11+ ) ( 15+19+ldots ) up to 10 terms A . 210 в. 201 ( c cdot 120 ) D. 102 | 11 |

1440 | Assertion For ( boldsymbol{n} in boldsymbol{N},(boldsymbol{n} !)^{3}boldsymbol{6},left(frac{boldsymbol{n}}{boldsymbol{3}}right)^{boldsymbol{n}}<boldsymbol{n} !left(frac{boldsymbol{n}}{2}right)^{boldsymbol{n}} ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion C. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect | 11 |

1441 | Is 402 a term of the sequence: ( 8,13,18,23, dots dots dots dots dots ) | 11 |

1442 | Ifx,y and z are pth, qth and rth terms respectively of an A.P. and also of a G.P., then xy – 2 yz – zx -y is equal to : (a) xyz (b) 0 (0) 1 (d) None of these | 11 |

1443 | Find the sum of first ( n ) term of an ( A . P ) is ( c n^{2}, ) which ( n t h ) term is ( t_{n}=5+6 n n in ) ( N ) A ( cdot frac{nleft(4 n^{2}-1right) c^{2}}{6} ) B. ( 3 n^{2}+8 n ) c. ( frac{nleft(4 n^{2}-1right) c^{2}}{3} ) D. ( frac{nleft(4 n^{2}+1right) c^{2}}{6} ) | 11 |

1444 | Find the ( n t h ) term and the sum of ( n ) terms of the series ( 1.2 .4+2.3 .5+ ) ( 3.4 .6+ldots ) | 11 |

1445 | If the ( (p+q)^{t h} ) term of a geometric series is ( mathrm{m} ) and the ( (p-q)^{t h} ) term is ( n ) then the ( p^{t h} ) term is ( mathbf{A} cdot(m n)^{1 / 2} ) в. ( m ) n ( c cdot m+n ) D. m-n | 11 |

1446 | Which term of the G.P.: ( 2,2 sqrt{2}, 4, dots ) is ( 128 ? ) | 11 |

1447 | ( 4,9,25,49, dots ) ( A cdot 64 ) B. 81 ( c cdot 12 ) D. 125 | 11 |

1448 | The sum of first 10 terms of the A.P. ( -2,1,4,7, ) is: A . 115 в. 230 ( c .1000 ) D. 620 | 11 |

1449 | The ( 4^{t h} ) term of an AP is zero.Prove that the ( 25^{t h} ) term is thrice its ( 11 t h ) term | 11 |

1450 | If ( |x|<1 ) then the coefficient of ( x^{5} ) in the expansion of ( frac{3 x}{(x-2)(x-1)} ) is A ( cdot frac{33}{32} ) в. ( -frac{33}{32} ) c. ( frac{31}{32} ) D. ( -frac{33}{34} ) | 11 |

1451 | Which term of the sequence 4,9,14,19 …, is ( 124 ? ) A . 25 B. 30 c. 15 D. 35 | 11 |

1452 | A student was given a piece of rope and told to cut it into two equal pieces, keep one piece, and pass the other piece to the next student. Each student was to repeat this process until every student in the class had exactly one piece of rope. Which of the following could be the fraction of the original rope that one of the students had? A ( cdot frac{1}{14} ) в. ( frac{1}{15} ) c. ( frac{1}{16} ) D. ( frac{1}{17} ) E ( frac{1}{18} ) | 11 |

1453 | Find four numbers in an ( A P ) whose sum is 20 and sum of whose square is 120 A .2,4,6,8 В. 2,-4,6,8 c. 2,4,7,8 D. 0,4,6,8 | 11 |

1454 | ( f S_{n} ) denotes the sum of ( n ) terms of an AP whose common differences is ( d ) show that ( boldsymbol{d}=boldsymbol{S}_{boldsymbol{n}}-boldsymbol{2} boldsymbol{S}_{boldsymbol{n}-1}+boldsymbol{S}_{boldsymbol{n}-boldsymbol{2}} ) | 11 |

1455 | Sum of the series ( 1+2.2+3.2^{2}+ ) ( 4.2^{3}+ldots+100.2^{99} ) is B. ( 99.2^{100}+1 ) D. ( 100.2^{100}-1 ) | 11 |

1456 | The angles of a quadrilateral are in ( boldsymbol{A} . boldsymbol{P} ) whose common difference is ( 10^{0}, ) then the first angle in ( A . P . ) is | 11 |

1457 | If ( t_{0}, ) represents ( n^{t h} ) term of an A.P. ( t_{2}+ ) ( t_{5}-t_{3}=10 ) and ( t_{2}+t_{9}=17, ) find its first term and its common difference. | 11 |

1458 | Prove that the sum of an odd number of terms in A.P. is equal to the middle term multiplied by the number of terms. | 11 |

1459 | The A.M. of ( a+2, a, 2-a ) is A . ( a ) в. ( frac{a+4}{3} ) c. ( frac{a-4}{3} ) D. | 11 |

1460 | Which term of the A.P. 5,12,19,26 is 145 A . 12 B. 18 ( c cdot 25 ) D. 21 | 11 |

1461 | It is known that ( sum_{r=1}^{infty} frac{1}{(2 r-1)^{2}}=frac{pi^{2}}{8} ) Then ( sum_{r=1}^{infty} frac{1}{r^{2}} ) is equal to A ( cdot frac{pi^{2}}{24} ) в. ( frac{pi^{2}}{3} ) c. ( frac{pi^{2}}{6} ) D. none of these | 11 |

1462 | 6. If n is a natural number such that n=p, 01. p2º2 • P3C3……..Pkºk and P1, P2, ….., Pk are istinct primes, then show that In nk ln2 (1984 – 2 Marks | 11 |

1463 | Find the first 3 terms of a G.P. if ( a=4 ) and ( r=2 ) | 11 |

1464 | ( sqrt{1+3+5+7+dots} ) | 11 |

1465 | What is the sum of all positive integers up to ( 1000, ) which are divisible by 5 and are not divisible by ( 2 ? ) A. 10,050 в. 5050 c. 5000 D. 50,000 | 11 |

1466 | In our number system the base is ten. If the base were changed to four you would count as follows: ( 1,2,3,10,11,12,13,20,21,22,23,30, dots ) The twentieth number would be: A . 20 B. 38 c. 44 D. 104 E. 110 | 11 |

1467 | If for any ( A . P, d=5 ) then ( t_{18}-t_{13} ) ( = ) ( mathbf{A} cdot mathbf{5} ) B. 20 c. 25 D. 30 | 11 |

1468 | If nth term of an A.P is ( (2 n+1), ) what is the sum of its first three terms? | 11 |

1469 | Given that ( x, y, z ) are positive real numbers such that ( x y z=32 . ) The minimum value of ( x^{2}+4 x y+4 y^{2}+ ) ( 2 z^{2} ) is equal to ( mathbf{A} cdot 64 ) в. 256 c. 96 D. 216 | 11 |

1470 | ( left(2^{2}+4^{2}+6^{2}+ldots ldots+20^{2}right)=? ) A . 77 В. 1155 ( c .1540 ) D. ( 385 times 385 ) | 11 |

1471 | The following consecutive terms ( frac{1}{1+sqrt{x}}, frac{2}{1-x}, frac{1}{1-sqrt{x}} ) of a series are in A. H.P в. G.P. c. A.P. D. None | 11 |

1472 | ( boldsymbol{A}=(mathbf{2}+mathbf{1})left(mathbf{2}^{2}+mathbf{1}right)left(mathbf{2}^{4}+right. ) 1) ( dots . .left(2^{2016}+1right) . ) The value of ( (boldsymbol{A}+mathbf{1})^{frac{1}{2016}} ) is ( mathbf{A} cdot mathbf{4} ) в. 2016 c. ( 2^{403} ) D. | 11 |

1473 | Write any two arithmetic progressions with common difference 3 | 11 |

1474 | 2, 7, 24, 77, 238, …. A . 721 B. 722 ( c cdot 723 ) D. 733 | 11 |

1475 | Find the AM between 20 and 26 A . 23 B. 22 ( c cdot 21 ) D . 24 | 11 |

1476 | Calculate the arithmetic mean, range, median and mode of the given data: ( mathbf{2}, mathbf{4}, mathbf{7}, mathbf{4}, mathbf{9}, mathbf{5}, mathbf{7}, mathbf{3}, mathbf{6}, mathbf{7} ) | 11 |

1477 | The ( 4^{t h} ) and ( 7^{t h} ) terms of a ( G P ) are ( frac{1}{27} ) and ( frac{1}{729} ) respectively. Find the sum of ( m ) terms of the ( G P ) | 11 |

1478 | Calculate ( 10^{t h} ) term of the infinite series ( 4,6,8, dots infty ) A . 18 B. 20 c. 22 D. 26 | 11 |

1479 | In the sequences ( 2,5,8, ldots ) upto 50 terms and ( 3,5,7, dots ) upto 60 terms, find the number of identical terms. | 11 |

1480 | ( frac{1^{3}}{1}+frac{1^{3}+2^{3}}{1+3}+frac{1^{3}+2^{3}+3^{3}}{1+3+5}+dots n ) terms ( = ) A. ( frac{nleft(2 n^{2}+9 n+13right)}{24} ) B. ( frac{nleft(2 n^{3}+9 n+13right)}{8} ) c. ( frac{nleft(n^{2}+9 n+13right)}{24} ) D. ( frac{nleft(n^{2}+9 n+13right)}{8} ) | 11 |

1481 | 29. If the 2nd, 5th and 9th terms of a non-consta “and 9n terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is: [JEEM 2016] (a) 1 WA AIN | 11 |

1482 | The sum of the first 7 terms of an A.P is 63 and the sum of its next 7 terms is 161. Find the 28 th term of this A.P. | 11 |

1483 | Identify whether the following sequence is a geometric sequence or not ( frac{1}{2}, frac{2}{3}, frac{3}{4}, frac{4}{5} ) | 11 |

1484 | Find the two missing numbers in the arithmetic mean between 14 and 50 A. 25 and 35 B. 26 and 38 c. 27 and 34 D. 21 and 32 | 11 |

1485 | The value of ( tan alpha+2 tan (2 alpha)+ ) ( 4 tan (4 alpha)+ldots+2^{n-1} tan left(2^{n-1} alpharight)+ ) ( 2^{n} cot left(2^{n} alpharight) ) is A ( cdot cot left(2^{n} alpharight) ) B ( cdot 2^{n} tan left(2^{n} alpharight) ) ( c ) D. ( cot alpha ) | 11 |

1486 | If the G.M of two numbers is 24 and their ( mathrm{H.M} ) is ( frac{72}{5}, ) Find the numbers | 11 |

1487 | A circle with area ( A_{1} ) is contained in the interior of a larger circle with area ( boldsymbol{A}_{mathbf{1}}+ ) ( A_{2} . ) If the radius of the larger circle is 3 and ( A_{1}, A_{2}, A_{1}+A_{2} ) are in A.P., what is the radius of the smaller circle? A ( cdot frac{sqrt{3}}{2} ) B. ( frac{2}{sqrt{3}} ) ( c cdot sqrt{3} ) D. ( frac{3}{2} ) | 11 |

1488 | Which of the following is not in the form of G.P.? A. ( 2+6+18+54+dots ) B. ( 3+12+48+192+ldots ) c. ( 1+4+7+10+dots ) D. ( 1+3+9+27+ldots ) | 11 |

1489 | ABCD is a square of length a, ( a epsilon N, a>1 ) Let ( L_{1}, L_{2}, L_{3}, dots ) be points on BC such that ( B L_{1}=L_{1} L_{2}=L_{2} L_{3}=ldots=1 ) and ( M_{1}, M_{2}, M_{3}, ldots ) be points on CD such that ( boldsymbol{C M}_{mathbf{1}}=boldsymbol{M}_{mathbf{1}} boldsymbol{M}_{mathbf{2}}=boldsymbol{M}_{mathbf{2}} boldsymbol{M}_{mathbf{3}}=ldots=mathbf{1} ) Then ( sum_{n=1}^{a-1}left(A L_{n}^{2}+L_{n} M_{n}^{2}right) ) is equal to A ( cdot frac{1}{2} a(a-1)^{2} ) в. ( frac{1}{2} a(a-1)(4 a-1) ) c. ( frac{1}{2} a(a-1)(2 a-1)(4 a-1) ) D. none of these | 11 |

1490 | The sum of ( n ) terms of the series ( 1+ ) ( (mathbf{1}+boldsymbol{a})+left(mathbf{1}+boldsymbol{a}+boldsymbol{a}^{2}right)+(mathbf{1}+boldsymbol{a}+ ) ( left.boldsymbol{a}^{2}+boldsymbol{a}^{3}right)+ldots . ) is A ( cdot frac{n}{1+a}-frac{1-a^{n}}{(1-a)^{2}} ) в. ( frac{n}{1-a}+frac{aleft(1-a^{n}right)}{(1-a)^{2}} ) c. ( frac{n}{1-a}+frac{aleft(1+a^{n}right)}{(1-a)^{2}} ) D. none of the above | 11 |

1491 | Which term of the A.P : 21, 18,15,…. is zero? | 11 |

1492 | If the ( 10^{t h} ) term and the ( 18^{t h} ) term of an A.P are 25 and 41 respectively then find the following ( ( a ) ) the ( 1^{s t} ) term and the common difference (b) the ( 38^{t h} ) term. | 11 |

1493 | If ( a_{1}, a_{2}, a_{3}, dots ) are in AP then ( a_{p}, a_{q}, a_{r} ) ( operatorname{are} operatorname{in} A P ) if ( p, q, r ) are in ( A cdot A P ) в. GP c. нр D. none of these | 11 |

1494 | The sum of ( 4^{text {th }} ) and ( 8^{text {th }} ) term of an A.P. is 24 and the sum of ( 6^{text {th }} ) and ( 10^{text {th }} ) term is 34. Find the first term and the common difference of A.P. | 11 |

1495 | If ( x, y ) and ( z ) are positive real numbers such that ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=boldsymbol{a} ) then ( – ) A ( cdot frac{1}{x}+frac{1}{y}+frac{1}{z} geq frac{9}{a} ) B. ( frac{1}{x}+frac{1}{y}+frac{1}{z}frac{8}{27} a^{3} ) D. ( (a-x)(a-y)(a-z)>a^{3} ) | 11 |

1496 | The arithmetic mean of first five natural numbers is A . 3 B. 4 c. 5 ( D ) | 11 |

1497 | If ( a^{2}, b^{2}, c^{2} ) are in A.P., then the following are also in A.P. ( frac{1}{b+c}, frac{1}{c+a}, frac{1}{a+b} ) | 11 |

1498 | Find the sum of the series ( 1^{2}+left(1^{2}+right. ) ( left.mathbf{2}^{2}right)+left(mathbf{1}^{2}+mathbf{2}^{2}+mathbf{3}^{2}right)+dots ) | 11 |

1499 | Let ( X ) be the set consisting of the first 2018 terms of the arithmetic progression ( 1,6,11,_{-} ) ( -, ) and ( Y b e ) the set consisting of the first 2018 terms of the arithmetic progression ( mathbf{9}, mathbf{1 6}, mathbf{2 3} ) . Then, the number of elements in the set XUY is ( ? ) | 11 |

1500 | There are four numbers in A.P. Their sum is 20 and sum of their squares is 120. Find largest of those numbers? | 11 |

1501 | Considre the sequence ( left(a_{n}right) n geq 0 ) given by the following relation: ( a_{0}=4, a_{1}= ) ( 22, ) and for all ( n geq 2, a_{n}=6 a_{n-1}- ) ( a_{n-2} . ) Prove that there exist sequences of positive integers ( left(x_{n}right) n geq ) ( mathbf{0},left(boldsymbol{y}_{n}right) boldsymbol{n} geq mathbf{0} ) such that ( boldsymbol{a}_{boldsymbol{n}}=frac{boldsymbol{y}_{boldsymbol{n}}^{boldsymbol{2}}+boldsymbol{7}}{boldsymbol{x}_{boldsymbol{n}}-boldsymbol{y}_{boldsymbol{n}}} ) for all ( n geq 0 ) | 11 |

1502 | 2. For 0<$<t/2, if x= cos2np , y = Àsin 2nd, 2= cos2n • sinan n=0 then: (a) xyz = xz + y (b) xyz = xy + z (c) xyz = x + y + z (d) xyz = yz + x n=0 n=0 (1993 – 2 Marks) | 11 |

1503 | The G.M. of n positive terms ( boldsymbol{x}_{1}, boldsymbol{x}_{2}, ldots . . . boldsymbol{x}_{n} ) is в. ( frac{1}{n}left(x_{1} times x_{2} times ldots . . times x_{n}right) ) C. ( left(x_{1} times x_{2} times ldots . times x_{n}right)^{1 / n} ) D. none of these | 11 |

1504 | Which term of the A.P. ( 17,16 frac{1}{5}, 15 frac{2}{5}, 14 frac{3}{5} dots . ) is first negative term? | 11 |

1505 | The third term of a geometric progression is ( 4 . ) The product of the first five terms is ( A cdot 4^{3} ) B . ( 4^{4} ) ( c cdot 4^{5} ) D. ( 4^{circ} ) | 11 |

1506 | can be one of the term in Arithmetic progression 4,7,10 A. 103 B. 123 c. 17 D. 99 | 11 |

1507 | For an ( A . P . ) if ( T_{n}=6 n+5 ) then find ( S_{n} ) | 11 |

1508 | f A.M and H.M between two positive numbers are 27 and 12 respectively find their G.M. | 11 |

1509 | f ( S_{1}, S_{2}, S_{3}, dots . . S_{r} ) are the sums of ( n ) terms of arithmetic series whose first terms are ( 1,2,3,4, dots ; ) and whose common differences are ( 1,3,5,7, dots ) find the value of ( boldsymbol{S}_{mathbf{1}}+boldsymbol{S}_{mathbf{2}}+boldsymbol{S}_{mathbf{3}}+ldots+ ) ( boldsymbol{S}_{boldsymbol{r}} ) | 11 |

1510 | Find the sum of the series whose nth term is : ( boldsymbol{n}^{3}-boldsymbol{3}^{n} ) | 11 |

1511 | Given the first two terms of an infinitely decreasing geometric progression ( sqrt{mathbf{3}}, frac{mathbf{2}}{sqrt{mathbf{3}}+mathbf{1}} ) Find the common ratio and the sum of the progression. | 11 |

1512 | If ( sum_{r=1}^{n}(r)(r+1)(2 r+3)=a n^{4}+ ) ( b n^{3}+c n^{2}+d n+e, ) then This question has multiple correct options A ( . a+c=b+d ) В. ( e=0 ) c. ( a, b-frac{2}{3}, c-1 ) are in A.P D. ( frac{c}{a} ) is an integers | 11 |

1513 | ( sum_{s=1}^{n}left{sum_{r=1}^{s} rright}=a n^{3}+b n^{2}+c n ) then find the value of ( a+b+c ) ( A ) B. ( c cdot 2 ) ( D ) | 11 |

1514 | If the sum of ( n ) terms of an ( A . P . ) is ( 3 n^{2}+5 n, ) then which of its term is 164 A . ( 26 t h ) B. 27th ( c .28 t h ) D. None of these | 11 |

1515 | An AP consists of 50 terms of which 3 rd term is 12 and the last term is ( 106 . ) Find the 29 th term. | 11 |

1516 | If in an ( A . P . ) the sum of ( m ) terms is equal to ( n ) and the sum of ( n ) terms is equal to ( m, ) then prove that sum of ( (m+n) ) terms is ( -(m+n) ) | 11 |

1517 | If ( sin ^{-1}left(x-frac{x^{2}}{2}+frac{x^{3}}{4}-dots dots inftyright)+ ) ( cos ^{-1}left(x^{2}-frac{x^{4}}{2}+frac{x^{6}}{4}-dots dotsright)= ) ( frac{pi}{2} ) and ( 0<x<sqrt{2} ) then ( x= ) ( A cdot frac{1}{2} ) B. ( c cdot-frac{1}{2} ) D. – | 11 |

1518 | ( mathbf{0 . 3}+mathbf{0 . 0 3 + 0 . 0 0 3 + 0 . 0 0 0 3}+ldots ldots . ) to terms. | 11 |

1519 | f ( p, q ) and ( r ) in ( A P ), then prove that ( (p+ ) ( mathbf{2} boldsymbol{q}-boldsymbol{r})(boldsymbol{2} boldsymbol{q}+boldsymbol{r}-boldsymbol{p})(boldsymbol{r}+boldsymbol{p}-boldsymbol{q})=boldsymbol{4} boldsymbol{p} boldsymbol{q} boldsymbol{r} ) | 11 |

1520 | The ( 5^{t h}, 8^{t h} ) and ( 11^{t h} ) terms of a G.P. are ( p, q & s ) respectively. Show that ( q^{2}=p s ) | 11 |

1521 | Let ( a_{1}, a ) and ( b_{1}, b_{2}, dots . ) be the arithmetic progressions such that ( a_{1}=25, b_{1}= ) 75 and ( a_{1} 00+b_{100} . ) The sum of the first one hundred terms of the progressions ( left(a_{1}+b_{1}right),left(a_{2}+b_{2}right), dots dots ) is ( A cdot O ) B. 100 ( c cdot 10,000 ) D. 5,05,000 | 11 |

1522 | If ( boldsymbol{x} epsilon boldsymbol{R}, ) the numbers ( boldsymbol{2}^{1+boldsymbol{x}}+ ) ( 2^{1-x}, frac{b}{2}, 36^{x}+36^{-x} ) form an A.P., then must lie in the interval ( mathbf{A} cdot[12, infty) ) B. ( [6, infty) ) ( c cdot(-infty, 6] ) ( mathbf{D} cdot[6,12] ) | 11 |

1523 | The series ( 1^{2}-2^{2}+3^{2}-4^{2}+ldots .+ ) ( 99^{2}-100^{2}= ) A . -5050 B. 5050 c. 11000 D. -11000 | 11 |

1524 | The number of real solution of the equations ( sin left(e^{x}right)=5^{x}+5^{-x} ) is ( mathbf{A} cdot mathbf{0} ) B. ( c cdot 2 ) D. infinitely many | 11 |

1525 | For the given A.P. ( 10,15,20,25, ldots . . ., ) find the common difference ‘ ( d ) | 11 |

1526 | If ( 6,6^{2}, 6^{3}, 6^{4} cdots 6^{n} ) be n observation then the quantity ( (sqrt{6})^{n+1} ) is called A . G.M в. Н.М c. А.М D. None of these | 11 |

1527 | How many terms of ( boldsymbol{A P}: 27,24,21, ldots . ) should be taken so that their sum is zero? What is the value of that last term | 11 |

1528 | Geometric mean of 3,9 and 27 is A . 18 B. 6 ( c .9 ) D. None of these | 11 |

1529 | The ( 4^{t h} & 7^{t h} ) of an are ( A . P ) are ( 17 & 23 ) respectively find o is. | 11 |

1530 | The ( n^{t h} ) term of an ( A . P ) is ( 7-4 n . ) Find its common difference. | 11 |

1531 | If sum of ( 3 r d ) and ( 8 t h ) terms of an ( A . P ) is 7 and sum of 7 th and 14 th terms is -3 then find the ( 10 t h ) term. | 11 |

1532 | In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying, e.g., a section of class I will plant 1 tree, a section of class II will plant 2 trees and so on till Class XII. There are three sections in each class. How many trees will be planted by the students | 11 |

1533 | [2005] 14. The sum of the series 11.1. * 4.2! 16.4! 64.6!” + + + – + …………….. …ad inf.is – A DI | 11 |

1534 | Which term of the progression 5,8,11 ( 14, dots ) is ( 320 ? ) A . 106 th B. 105 th c. 107 th D. 104 th | 11 |

1535 | is 0 a term of the ( A P: 31,28,25, ldots ? ) If true then enter 1 and if false then enter 0 | 11 |

1536 | Find the first term of a G.P. in which ( S_{8}=510 ) and ( r=2 ) | 11 |

1537 | If the arithmetic mean of n numbers of a series is ( bar{x} ) and the sum of the first ( (n- ) 1) numbers is ( k ), then the nth number is ( A cdot n+k ) B . ( n bar{x}+k ) c. ( n bar{x}-k ) D. n-k | 11 |

1538 | Arithmetic progression whose nth term is ( 3 n-2 ) is A ( .1,4,7,10 ldots ) в. ( 1,3,7,9 ldots ) c. ( 1,2,7,11 ldots ) D. ( 2,4,6,8 dots ) | 11 |

1539 | Find 9 th term of the ( A . P frac{3}{4}, frac{5}{4}, frac{7}{4}, frac{9}{4} ) | 11 |

1540 | The sum ( 1+frac{2}{x}+frac{4}{x^{2}}+frac{8}{x^{3}}+ ) ( ldots(u p t o infty), x neq 0, ) is finite if A ( cdot|x|2 ) c ( cdot|x|<1 ) D . ( 2|x|<1 ) | 11 |

1541 | Find the sum to ( n ) terms of the ( A . P . ) whose ( k^{t h} ) term is ( 5 k+1 ) | 11 |

1542 | 17. The sum of series 1 1 – ….. upto infinity is [2007] (a) e 2 (6 e 2 (c) e2 (2) e1 | 11 |

1543 | Show that in an arithmetical progression ( a_{1}, a_{2}, a_{3} ) ( boldsymbol{S}=boldsymbol{a}_{1}^{2}-boldsymbol{a}_{2}^{2}+boldsymbol{a}_{3}^{2}-boldsymbol{a}_{4}^{2}+_{–} ) ( boldsymbol{a}_{2 k}^{2} ) ( =frac{k}{2 k-1}left(a_{1}^{2}-a_{2 k}^{2}right) ) | 11 |

1544 | Between the number 1 and ( 31, m ) means are inserted so that the ratio of ( 7^{t h} ) and ( (m-1)^{t h} ) means is ( 5: 9, ) then the value of ( m ) is A . 14 B. 10 ( c cdot 7 ) D. 3 | 11 |

1545 | For what value of ( n ) are the ( n^{t h} ) terms of the following two A.P’s the same? (i) ( 1,7,13,19, dots ) (ii) ( 69,68,67, dots ) | 11 |

1546 | The G.M. of numbers 4,5,10,20,25 is A . 12.8 B. 10 ( c .7 .8 ) D. none of these | 11 |

1547 | The mean of ( 12,65,84,75, a ) is 50 Find a | 11 |

1548 | Find the sum of ( 2 n ) terms of the series ( mathbf{5}^{mathbf{3}}+mathbf{4 . 6}^{mathbf{3}}+mathbf{7}^{mathbf{3}}+mathbf{7}^{mathbf{3}}+mathbf{4 . 8}^{mathbf{3}}+mathbf{9}^{mathbf{3}}+ ) ( 4.10^{3}+ldots ) | 11 |

1549 | Find the arithmetic mean between 9 and 19 | 11 |

1550 | For the following arithmetic progression write the first term and common difference (i) ( frac{1}{3}, frac{5}{3}, frac{9}{3}, frac{13}{3}, dots ) (ii) ( 0.6,1.7,2.8,3.9, dots ) | 11 |

1551 | The sum of ( 6^{t h} ) term in the geometric ( operatorname{series} 4,12,36 dots ) is A . 1456 в. 2456 ( c .3456 ) D. 4456 | 11 |

1552 | ( boldsymbol{a}_{boldsymbol{n}}=frac{boldsymbol{n}}{boldsymbol{n}+mathbf{1}} ) | 11 |

1553 | f ( a, b, c ) are positive real numbers such that ( a b^{2} c^{3}=64 ) then minimum value of ( left(frac{1}{a}+frac{2}{b}+frac{3}{c}right) ) is equal to ( A cdot 6 ) B. 3 ( c cdot 2 ) D. None of these | 11 |

1554 | 1. If the first and the (2n-1)st terms of an A.P., a G.P. ?-1)st terms of an A.P., a G.P. and an H.P. are equal and their n-th terms are a, b and crespect then (1988 – 2 Marks) (a) a=b=c (b) a 2 b2c (c) a+c=b (d) ac- 62 = 0. | 11 |

1555 | The arithmetic mean of numbers ( a, b, c, d, e ) is ( M . ) What is the value of ( (boldsymbol{a}-boldsymbol{M})+(boldsymbol{b}-boldsymbol{M})+(boldsymbol{c}-boldsymbol{M})+(boldsymbol{d}- ) ( M)+(e-M) ? ) ( mathbf{A} cdot M ) B. ( a+b+c+d+e ) ( c cdot 0 ) ( D .5 M ) | 11 |

1556 | Find out which of the following sequences are arithmetic progressions. For those which are arithmetic progressions, find out the common difference.10 ( +10+2^{5}, 10+2^{6}, 10+ ) ( mathbf{2}^{7}, ldots . . ) | 11 |

1557 | Find the sum of ( 2,7,12, ldots ) to 10 terms A. 160 в. 245 ( c cdot 290 ) D. 300 | 11 |

1558 | In the four numbers first three are in G.P. and last three are in A.P. whose common difference is ( 6 . ) If the first and last numbers are same, then first will be | 11 |

1559 | Find the ( 15^{t h} ) term of the G.P. 3,12,48 ( 192, ldots ) A ( cdot 3 times 4^{15} ) B . ( 3 times 4^{14} ) ( mathrm{c} cdot 3 times 4^{16} ) D. ( 3^{15} ) | 11 |

1560 | Six positive number are in G.P., such that the ( r ) product is ( 1000 . ) If the fourth term is ( 1, ) then the last term is A. 1000 в. 100 c. ( frac{1}{100} ) D. ( frac{1}{1000} ) | 11 |

1561 | Find the eleventh term of the A.P. ( 7,13,19,25, dots dots ) | 11 |

1562 | Find the 5 th term of the sequence ( 1, sqrt{2}, 2 ldots ) A. B. 2 ( c cdot 3 ) ( D ) | 11 |

1563 | If the ( 10^{t h} ) term of an A.P. is 52 and the ( 17^{t h} ) term is 20 more than the ( 13^{t h} ) term, find the A.P. | 11 |

1564 | Find the sum of 90 terms of A.P ( 4,8,12, dots ) | 11 |

1565 | The least value of ( frac{1}{2}+frac{2}{3} operatorname{cosec}^{2} theta+ ) ( frac{3}{8} sec ^{2} theta ) is A ( cdot frac{13}{24} ) в. ( frac{61}{48} ) c. ( frac{61}{25} ) D. ( frac{61}{24} ) | 11 |

1566 | 3. The value of 21/4.41/8.81/16 … oo is (a) 1 (6) 2 (c) 3/2 [200 (d) 4 D:41. | 11 |

1567 | If the first term of a geometric series is ( 2, ) and its third term is ( 8, ) how many digits are there in the ( 40^{t h} ) term of the series, if the common ratio of the sequence is positive? A . 10 B. 11 c. 12 D. 13 E. 14 | 11 |

1568 | bn = 1 -an, then find the least natural number such that bn > a, & n>no. (2006 – 6M) | 11 |

1569 | If the 3 rd and the sth term of an AP are 4 and – 8 respectively, which term of this AP is zero? | 11 |

1570 | If ( a, b, c ) are three positive real number then show : ( frac{a}{b+c}+frac{b}{a+c}+ ) ( frac{c}{a+b} geq frac{3}{2} ) | 11 |

1571 | ( ln ) an A.P., prove that ( boldsymbol{d}=boldsymbol{S}_{boldsymbol{n}}-boldsymbol{2} boldsymbol{S}_{boldsymbol{n}-boldsymbol{1}}+ ) ( boldsymbol{S}_{boldsymbol{n}-boldsymbol{2}} ) | 11 |

1572 | An ( A P ) consists of 21 terms. The sum of the three terms in the middle is ( 129 & ) of the last three is 237 . Find ( A P ) | 11 |

1573 | Assertion(A): ( A, B, C ) are positive angles such that ( A+B+C=7 pi ), then maximum value of ( cot A cot B cot C= ) ( frac{1}{3 sqrt{3}} ) Reason(R): ( A . M . geq G . M ) A. A is true, R is false B. Both A and R are false c. A is false, R is true D. Both A and R are true | 11 |

1574 | The ( n ) th term of an A.P is ( 6 n+2 ). Find the common difference. | 11 |

1575 | State True or False. ( f(a, b, c, d ) are four positive real numbers such that ( a b c d=1 ), then ( (1+a)(1+b)(1+c)(1+d) geq 16 ) A. True B. False | 11 |

1576 | If ( 6 t h ) term of a G.P is ( -1 / 32, ) and 9 th term is ( 1 / 256, ) then the 11 th term is A . 1024 B. 1/1024 c. ( 1 / 256 ) D. ( 1 / 512 ) | 11 |

1577 | Find the sum of first 15 terms of an A.P. whose 5 th and 9 th terms are 26 and 42 respectively | 11 |

1578 | Choose the correct option. This question has multiple correct options ( mathbf{A} cdot T_{n+2}-T_{n-1} ) is divisible by 2 ( ^{mathbf{B}} cdot frac{T_{n+1}+T_{n-1}}{T_{n}}=2 r ) C. Neither (A) nor (B) D. None of the above | 11 |

1579 | If 25 is the arithmetic mean between ( x ) and ( 46, ) then find ( x ) ( A cdot 2 ) B. 4 ( c cdot 8 ) ( D cdot 16 ) | 11 |

1580 | The ( 4^{t h} ) term of A.P. is 22 and ( 15^{t h} ) term is ( 66 . ) Find the first term and the common difference. Hence find the sum of the series to 8 terms. | 11 |

1581 | Write first four terms of the ( A P, ) when the first term a and the common difference d are given as follows: ( (i) a=10 d=10 ) (ii) ( a=-2, d=0-3 ) (iii) ( a=4, d=-3 ) (iv) ( a=-1, d=frac{1}{2} ) ( (v) a=-1, d=frac{1}{2} ) | 11 |

1582 | Find the next one term of the following sequence: ( 4,7,10,13, dots ) | 11 |

1583 | The eighth term of an AP is half its second term and the eleventh term exceeds one third of its fourth term by 1 Find the 15 th term. | 11 |

1584 | If ( n>0, ) prove that ( 2^{n}>1+n sqrt{2^{n-1}} ) | 11 |

1585 | If in a G.P., 5th term and the 12th term are 9 and ( frac{1}{243} ) respectively, find the 9 th term of G.P. A ( cdot frac{1}{7} ) B. ( frac{1}{8} ) ( c cdot frac{1}{9} ) D. ( frac{1}{81} ) | 11 |

1586 | Find the ( 10^{t h} ) term from end for the A.P. ( mathbf{3 . 6 . 9 . 1 2} ldots ldots .300 ) | 11 |

1587 | An Arithmetic Series is defined as ( boldsymbol{f}(boldsymbol{n})=boldsymbol{a}+(boldsymbol{n}-mathbf{1}) boldsymbol{d}, boldsymbol{n} in boldsymbol{N}, ) prove that A.M. of ( boldsymbol{f}(mathbf{1}) ) and ( boldsymbol{f}(boldsymbol{2 n}-mathbf{1}) ) is ( boldsymbol{f}(boldsymbol{n}) ) | 11 |

1588 | Find the number of terms of the A.P. ( -12,-9,-6,-ldots, 21 . ) If 1 is added to each term of this A.P., then find the sum of all terms of the A.P. thus obtained. | 11 |

1589 | If first, second and last terms of an A.P. are ( a, b ) and ( c ) respectively. Then the number of terms = A ( cdot frac{a+b-2 c}{a-c} ) в. ( frac{c+a-2 b}{c-b} ) c. ( frac{b+c-2 a}{b-a} ) D. ( frac{a+b+2 c}{2(b-c)} ) | 11 |

1590 | What is the common difference of an AP in which the ratio of the product of the first and fourth term to the product of the second and third term is ( 2: 3 ? ) It is given that the sum of the four terms is 20 A . 3 B. 4 c. 1 ( D ) | 11 |

1591 | If the sum of first ( n ) term of A.P is ( frac{1}{2}left(3 n^{2}+7 nright) ) then its ( n^{t} h ) term. hence ( 20^{t} h ) term | 11 |

1592 | If the ( 5^{t h} ) term of an ( A P ) is three times the first term then prove that the ( 7^{t h} ) term is twice the third term. | 11 |

1593 | The third term of an A.P is 7 and the seventh term exceeds three times the third term by 2. Find the first term, the common difference and the sum of first 20 terms. | 11 |

1594 | There are ( n ) AM’s between ( 1 & 31 ) such that ( 7 t h ) mean: ( (n-1)^{t h} ) mean ( =5: 9 ) then find the value of ( n ) | 11 |

1595 | | S2 °o 1, 0, – 1 4. ( UV0,1 Let T be the rth term of an A.P., for r=1,2,3, positive integers m, n we have 1.P., for r=1,2,3, …. If for some — and 7, – I then quals 1998-2 T = — annm (1998 – 2 Marks) (a) then I’mn equals (b) 1+1 (c) (c) 1 (d) o mn m n | 11 |

1596 | There are two sections ( A ) and ( B ) of a class consisting of 36 and 44 students respectively. If the average weight of section ( boldsymbol{A} ) is ( mathbf{4 0} ) kg and that of section ( boldsymbol{B} ) is ( 35 mathrm{kg} ), find the average weight of the whole class? | 11 |

1597 | State whether the given list of numbers is an arithmetic progression or not. ( a) 13,20,27,34, dots ) ( boldsymbol{b}) boldsymbol{6}, boldsymbol{1} boldsymbol{6}, boldsymbol{2} boldsymbol{6}, boldsymbol{3} 5, ldots ) | 11 |

1598 | ( boldsymbol{n}^{2}+mathbf{2}^{boldsymbol{n}} ) | 11 |

1599 | Find the 6 th term from the end of the A.P. ( 17,14,11, dots dots,-40 ? ) | 11 |

1600 | Find the next term of the sequence: ( mathbf{0 . 5}, mathbf{2}, mathbf{3 . 5}, mathbf{5}, dots dots dots ) A . 5.5 B. 6 ( c .6 .5 ) ( D ) | 11 |

1601 | Find the sum of the series whose nth term is: ( 2 n^{3}+3 n^{2}-1 ) | 11 |

1602 | [ begin{aligned} operatorname{Let} x &=111 ldots 11(20 text { digits }) \ y &=333 ldots 33(10 text { digits }) \ text { and } z &=222 ldots 22(10 text { digits }) end{aligned} ] The value of ( frac{boldsymbol{x}-boldsymbol{y}^{2}}{boldsymbol{z}} ) | 11 |

1603 | ( 1+frac{3}{2}+frac{5}{4}+frac{7}{8}+ldots . . ) n terms | 11 |

1604 | In an acute angled triangle ( A B C ), the minimum value of ( tan ^{n} A+tan ^{n} B+ ) ( tan ^{n} C . ) is ( (text { When } n epsilon N, n>1) ) A ( cdot frac{n}{3} ) B. ( 3^{n} ) c. ( frac{n}{3}^{+1} ) D. ( _{3} frac{n}{2}-1 ) | 11 |

1605 | The value of ( sum_{n=1}^{9999} frac{1}{(sqrt{n}+sqrt{n+1})(sqrt[4]{n}+sqrt[4]{n+1})} ) is ( mathbf{A} cdot mathbf{9} ) B. 99 ( c .999 ) D. 9999 | 11 |

1606 | Let a 1, 02, az, ….. be in harmonic progression w and a20 = 25. The least positive integer n an <0 is (2) 22 (6) 23 (c) 24 (d) 25 harmonic progression with a = 5 Positive integer n for which (2012) 10 11 1o | 11 |

1607 | ( frac{1}{2}+frac{1}{4}+frac{1}{6}+frac{1}{8}+ldots . ) is a A. sequence B. series c. term D. constant | 11 |

1608 | Adding all the terms in a sequence is called A. sequence B. series c. term D. constant | 11 |

1609 | If the coefficient of second, third and fourth terms in the expansion if ( (1+ ) ( x)^{2 n} ) are in A.P, the ( 2 n^{2}-3 n ) is equal to A. -7 B. 14 ( c .6 ) D. -6 | 11 |

1610 | In a geometric series, the first term ( =a ) common ratio ( =r . ) If ( S_{n} ) denotes the sum of the ( n ) terms and ( U_{n}=sum_{n=1}^{n} S_{n} ) then ( r S_{n}+(1-r) U_{n} ) equals to ( mathbf{A} cdot mathbf{0} ) B. ( c cdot n a ) D. ( n a ) | 11 |

1611 | The least value of ( 2 log _{100} a- ) ( log _{a} 0.0001, a>1 ) is ( A cdot 2 ) B. 3 ( c cdot 4 ) D. none of these | 11 |

1612 | If the sum of 5 terms of an ( A . P . ) is same as the sum of its 11 terms then sum of ( mathbf{1 6} ) is A. 0 B . 16 ( c .-16 ) D. 32 | 11 |

1613 | What is the fifth term of the arithmetic sequence ( 2,_{-}, 8, ldots, ldots ? ) ( mathbf{A} cdot mathbf{5} ) B. 11 c. 13 D. 14 E . 15 | 11 |

1614 | If the sum of four consecutive even numbers is ( 532, ) find the numbers. | 11 |

1615 | How many terms of the sequence ( sqrt{3}, 3,3 sqrt{3}, ldots ) must be taken to get the ( operatorname{sum} 39+13 sqrt{3} ? ) | 11 |

1616 | laluleu-l lui, sulmon 24. ( The sum of first 20 terms of the sequence 0.7, 0.77,0.777,….., (JEEM 2013] (@) 57079-10-20) (C) 57 79+10-20) () (99 –10-20) (d) ?(99+10-20) | 11 |

1617 | If ( a^{2}+b^{2}+c^{2}=1, x^{2}+y^{2}+z^{2}=1 ) where ( a, b, c, x, y, z ) are real, prove that ( boldsymbol{a} boldsymbol{x}+boldsymbol{b} boldsymbol{y}+boldsymbol{c} boldsymbol{z} leq mathbf{1} ) | 11 |

1618 | Find the sum of ( 1+4+7+10+dots dots ) to 22 terms | 11 |

1619 | Find ( S_{n}, ) the sum of the first ( n ) terms, for the following geometric series. ( a_{1}= ) ( mathbf{1 2 0}, boldsymbol{a}_{5}=mathbf{1}, boldsymbol{r}=-mathbf{2} ) A .20 .66 B. 40.66 c. 80.66 D. 100.66 | 11 |

1620 | Find the sequence if, ( boldsymbol{T}_{boldsymbol{n}}=mathbf{5} boldsymbol{n}+mathbf{1} ) | 11 |

1621 | Let ( a_{1}, a_{2}, a_{3} ) and ( a_{4} ) be in AP. If ( a_{1}+ ) ( a_{4}=10 ) and ( a_{2} cdot a_{3}=24, ) then the least term of them is A . 1 B. 2 ( c .3 ) D. 4 E. 5 | 11 |

1622 | If ( a^{2}, b^{2}, c^{2} ) are in A.P. then the following are also in A.P. True or False? If true write 1 otherwise write 0 ( frac{boldsymbol{a}}{boldsymbol{b}+boldsymbol{c}}, frac{boldsymbol{b}}{boldsymbol{c}+boldsymbol{a}}, frac{boldsymbol{c}}{boldsymbol{a}+boldsymbol{b}} ) | 11 |

1623 | For which sequence below can we use the formula for the general term of a geometric sequence? A. ( 1,3,5,7,9, ldots ) . в. ( 2,4,6,8,10 ldots ) c. ( 4,8,16,32,64 dots ) D. 1,-1,3,-2,4 | 11 |

1624 | 1,3,9,27,81 is a A. geometric sequence B. arithmetic progression C. harmonic sequence D. geometric series | 11 |

1625 | 10. rk) et r Consider an infinite geometric series with first term a and common ratio r. If its sum is 4 and the second term is 3/4, then (2000) G 4 3 (a) a=5,r== 7 7 31 (d) a=3,r= (b) a = 2,r=3 | 11 |

1626 | How many terms are need in the A.P. ( 24,20,16 ldots . . ) to make the sum ( 72 ? ) | 11 |

1627 | If numbers ( a, b ) and ( c ) are in ( A P, ) then A. ( b-a=c-b ) B. ( b+a=c+b ) c. ( a-c=b-d ) D. None of these | 11 |

1628 | Find the geometric mean of the following pairs of numbers: ( a^{3} b ) and ( a b^{3} ) ( mathbf{A} cdot a^{2} b^{2} ) B. ( a b ) ( c cdot a b^{2} ) D. ( a^{2} b ) | 11 |

1629 | If the ( 7^{t h} ) and ( 13^{t h} ) term an Arithmetic progression are 34 and 64 respectively. then the common difference is, ( A cdot 5 ) B. 10 c. 17 D. 32 | 11 |

1630 | If ( A ) is the area and ( 2 s ) the sum of three sides of a triangle, then ( ^{mathrm{A}} cdot_{A} leq frac{s^{1.5}}{3 sqrt{3}} ) в. ( _{A} leq frac{s^{1.5}}{2} ) ( ^{mathrm{c}} cdot_{A}>frac{s^{1.5}}{sqrt{3}} ) D. None of these | 11 |

1631 | Find the sum of 100 terms of the series ( mathbf{1}(mathbf{3})+mathbf{3}(mathbf{5})+mathbf{5}(mathbf{7})+cdots cdots ) A. 1353300 B. 1353400 c. 1353200 D. 1353100 | 11 |

1632 | Check if the series is an ( A P ). Find the common difference ( d . ) Also, find the next three terms. ( -10,-6,-2,2 dots dots ) A. It is an ( A P ) and ( d=4, ) other terms 6,10,14 B. It is an ( A P ) and ( d=frac{3}{5} ), other terms 5,10,15 c. It is not an ( A P ), other terms 3,4,5 D. None of these | 11 |

1633 | If the ratio of sum of ( n ) terms of two A.P.’s is ( (3 n+8):(7 n+15), ) then the ratio of ( 12^{t h} ) terms is ( mathbf{A} cdot 16: 7 ) B. 7: 16 c. 7: 12 D. 12: 5 | 11 |

1634 | Write the arithmetic progression when first term ( a ) and common difference ( d ) are as follows: ( boldsymbol{a}=-mathbf{1} ; boldsymbol{d}=frac{mathbf{1}}{mathbf{2}} ) | 11 |

1635 | The function ( 3 .(2)^{n-1} ) will follow which sequences? ( mathbf{A} cdot 2,6,12,24 ) в. ( 2,4,12,24 . ) c. 0,6,12,24 D. 3,6,12,24 | 11 |

1636 | Find the four numbers in A.P, whose sum is 50 and in which the greatest number is four times the least. | 11 |

1637 | The numbers ( a, b ) and ( c ) are between 2 and 18 such that (i) their sum is 25 ( (i i) ) the numbers ( 2, a, b ) are in A.P ( (i i i) ) the numbers ( b, c, 18 ) are consecutive terms of a G.P. If ( a, b, c ) are the roots of ( x^{3}+q x^{2}+ ) ( r x+s=0 ) then the value of ( ^{prime} r^{prime} ) is A. 184 в. 196 c. 220 D. 224 | 11 |

1638 | In a given ( A P ., ) if the p term is ‘q’and the ( q^{t h} ) tent is ‘p’, then its ( n^{t h} ) term is ( mathbf{A} cdot p+q-n ) B . ( p+q+n ) C ( . p-q+n ) D. ( p-q-n ) | 11 |

1639 | Find the sum to ( n ) terms of the A.P ( mathbf{5}, mathbf{2},-mathbf{1},-mathbf{4},-mathbf{7}, dots . ) | 11 |

1640 | Find ( 4 t h ) and ( 8 t h ) terms of the G.P. ( mathbf{0 . 0 0 8}, mathbf{0 . 0 4}, mathbf{0 . 2}, dots dots dots ) | 11 |

1641 | Moses deposited Rs. 850 into the bank in July. From July to December, the amount of money he deposited into the bank increase by ( 25 % ) per month. Whats the total amount of money in his account after December? A . 6570 в. 7570 ( c .8570 ) D. 9570 | 11 |

1642 | Find five terms in A.P. whose sum is ( 12 frac{1}{2} ) and the ratio of first to the last term is 2: 3 | 11 |

1643 | Find the sum of the following geometric series ( 1+frac{1}{3}+frac{1}{9}+ldots ldots ldots . ) upto ( infty ) | 11 |

1644 | If ( a>1, b>1, ) then the minimum value of ( log _{b} a+log _{a} b ) is A . B. 1 ( c cdot 2 ) D. none of these | 11 |

1645 | The first term of an arithmetic progression ( a_{1}, a_{2}, a_{3}, dots ) is equal to unity. At what value of the difference of the progression is ( a_{1} a_{3}+a_{2} a_{3} ) at a minimum? | 11 |

1646 | If the mean of ( x, x+2, x+4, x+8 ) is 20 find ( x ) | 11 |

1647 | The ( 8^{t h} ) term of an AP is 37 and its ( 12^{t h} ) term is ( 57 . ) Find the AP. | 11 |

1648 | The arithmetic mean (average) of the first ( n ) positive integers is A ( cdot frac{n}{2} ) в. ( frac{n^{2}}{2} ) c. ( n ) D. ( frac{n-1}{2} ) E ( cdot frac{n+1}{2} ) | 11 |

1649 | Insert five numbers between 4 and 8 so that the resulting sequence is an ( boldsymbol{A} . boldsymbol{P} ) | 11 |

1650 | Mid point ( boldsymbol{A}(mathbf{0}, mathbf{0}) ) and ( boldsymbol{B}(mathbf{1 0 2 4}, mathbf{2 0 4 8}) ) is ( A_{1} ) and mid point of ( A_{1} ) and ( B ) is ( A_{2} ) and so on. Coordinates of ( boldsymbol{A}_{mathbf{1 0}} ) are: A. (1022,2044) ) (1022,2444) B. (1025,2050) c. (1023,2046) D. (1,2) | 11 |

1651 | The sum of the series ( frac{1}{2}+frac{1}{3}+frac{1}{6}+ ) to 9 terms be ( k . ) Find ( -2 k ) | 11 |

1652 | Identify whether the following sequence is a geometric sequence or not -3,9,-27,81 | 11 |

1653 | Let ( s_{1}(n) ) be the sum of the first n terms of the arithmetic progression ( 8,12,16, ) and let ( s_{2}(n) ) be the sum of the first ( n ) terms of arithmetic progression 17,19,21 …. If for some value of ( n, s_{1}(n)=s_{2}(n) ) then this common sum is A. not uniquely determinable B. 260 ( c cdot 216 ) D. 200 | 11 |

1654 | For an ( A . P, ) if ( T_{1}=22, T_{n}=-11 ) and ( boldsymbol{S}_{boldsymbol{n}}=boldsymbol{6 6}, ) then find ( boldsymbol{n} ) | 11 |

1655 | Find ( x ), if the given numbers are in A.P. 5 ( (x-1), 0 ) | 11 |

1656 | The ( 4^{t h} ) and ( 10^{t h} ) terms of an AP are 13 and 25 respectively. Find the first term and the common difference of the AP. Also, find its ( 17^{t h} ) term. | 11 |

1657 | The first term in an arithmetic sequence is -5 and the second term is ( -3 . ) What is the 50 th term? (Recall that in an arithmetic sequence, the difference between successive terms is constant) ( A cdot 87 ) B . 78 ( c cdot 74 ) D. 93 | 11 |

1658 | The arithmetic mean of the series ( 1,2,2^{2}, dots 2^{n-1} ) is A ( cdot frac{2^{n}}{n} ) B. ( frac{left(2^{n}-1right)}{n} ) ( ^{mathrm{C}} cdot frac{left(2^{n+1}right)}{n} ) D. None of these | 11 |

1659 | Prove ( : 1^{2}+left(1^{2}+2^{2}right)+ ) ( left(1^{2}+2^{2}+3^{2}right)+ldots ) upto ( n ) terms ( = ) ( frac{n(n+1)^{2}(n+2)}{12} ) | 11 |

1660 | Find the common difference of the A.P. and write the next two terms. ( mathbf{5 1}, mathbf{5 9}, mathbf{6 7}, mathbf{7 5}, dots dots ) | 11 |

1661 | ( left(frac{1}{4}+frac{1}{4^{2}}+frac{1}{4^{3}}—-+frac{1}{4^{n-1}}right) ) | 11 |

1662 | The sum of the infinite series ( 1+frac{1+2}{2 !}+frac{1+2+2^{2}}{3 !}+frac{1+2+2^{2}+2^{3}}{4 !} ) is ( e^{y}-e^{x} ) Find ( x+y^{2} ) | 11 |

1663 | If the ( p^{t h}, q^{t h}, r^{r h} ) terms of a G.P. be ( a, b, c ) respectively, then ( mathbf{A} cdot a^{q-r} b^{r-p} c^{p-q}=1 ) B . ( a^{q-r} b^{r-p} c^{p-q}=-1 ) ( mathbf{c} cdot a^{q-r} b^{r-p} c^{p-q}=0 ) D. none of these | 11 |

1664 | Consider two arithmetic series: [ begin{array}{l} boldsymbol{A}_{1}: mathbf{2}+mathbf{9}+mathbf{1 6}+mathbf{2 3}+ldots ldots ldots+mathbf{2 0} \ boldsymbol{A}_{mathbf{2}}: mathbf{5}+mathbf{9}+mathbf{1 3}+mathbf{1 7}+ldots ldots ldots+mathbf{1 6} end{array} ] then the number of terms common to the two series is ( A cdot 6 ) B. 8 c. 10 D. 12 | 11 |

1665 | If ( |a|<1 ) and ( |b|<1 ) then ( boldsymbol{S}=mathbf{1}+(mathbf{1}+boldsymbol{a}) boldsymbol{b}+left(mathbf{1}+boldsymbol{a}+boldsymbol{a}^{2}right) boldsymbol{b}^{2}+dots ) ( = ) A ( cdot frac{1}{(1-b)(1-a b)} ) B. ( frac{1}{(1+b)(1-a b)} ) c. ( frac{1}{(1-b)(1+a b)} ) D. ( frac{1}{(1+b)(1+a b)} ) | 11 |

1666 | The first four terms of an ( A . P . ) whose first term is 2 and the common difference is 2 are: B. 2,4,8,16 ( mathrm{c} .2,4,6,8 ) D. 2,5,8,11 | 11 |

1667 | ( operatorname{Sum} 3,-4, frac{16}{3}, dots ) to ( 2 n ) terms. | 11 |

1668 | Evaluate ( 1+i^{2}+i^{4}+i^{6}+ldots+i^{2 n} ) | 11 |

1669 | Let ( boldsymbol{f}(boldsymbol{x})=log left(1+boldsymbol{x}^{2}right) ) and ( boldsymbol{A} ) be a constant such that ( frac{|boldsymbol{f}(boldsymbol{x})-boldsymbol{f}(boldsymbol{y})|}{|boldsymbol{x}-boldsymbol{y}|} leq boldsymbol{A} ) for all ( x, y ) real and ( x neq y . ) Then the least possible value of ( boldsymbol{A} ) is A. Equal to 1 B. Bigger than 1 but less than 2 c. Bigger than 0 but less than 1 D. Bigger than 2 | 11 |

1670 | ( 25 t h ) term of the ( A . P: 5, frac{5}{2}, 0,-frac{5}{2}, ldots ) | 11 |

1671 | For a sequence if ( S_{n}=frac{4^{n}-3^{n}}{3^{n}} ) find the nth term hence show that if it is a G.P | 11 |

1672 | The sequence ( -10,-6,-2,2, dots dots ) ( mathbf{A} cdot ) is an ( A cdot P ., ) Reasons ( d=-16 ) B. is an ( A . P ., ) Reasons ( d=4 ) C. is an ( A ).P., Reasons ( d=-4 ) D. is not an ( A . P ) | 11 |

1673 | If the sum of the ( 3^{r d} ) and the ( 8^{t h} ) terms of an AP is 7 and the sum of the ( 7^{t h} ) and the ( 14^{t h} ) term is ( -3, ) find the ( 10^{t h} ) term. | 11 |

1674 | If the arithmetic mean of ( n ) numbers of a series is ( bar{x} ) and sum of the first ( (n- ) 1) numbers is ( k, ) then which one of the following is the nth number of the series ( ? ) A . ( bar{x}-n k ) В . ( n bar{x}-k ) c. ( k bar{x}-n ) D. ( n k bar{x} ) | 11 |

1675 | f ( a, b, c ) are in A.P then ( aleft(frac{1}{b}+frac{1}{c}right), bleft(frac{1}{c}+frac{1}{a}right), cleft(frac{1}{a}+frac{1}{b}right) ) are in A. ( A . P . ) в. G.P. c . ( H . P ) D. A.G.P | 11 |

1676 | If ( frac{a^{n}+b^{n}}{a^{n-1}+b^{n-1}} ) is the A.M. between ( a ) and ( b, ) then find the value of ( n ) | 11 |

1677 | Which term of the G.P.: ( sqrt{mathbf{3}}, mathbf{3}, mathbf{3} sqrt{mathbf{3}}, ldots ) is 729? | 11 |

1678 | Find the sum of series ( 1+(2)(3)+ ) ( (4)+(5)(6)+7+ldots . ) upto 50 term | 11 |

1679 | If the number ( 3 k+4,7 k+1 ) and ( 12 k- ) 5 are in A.P., then the value of ( k ) is ( A cdot 2 ) B. 3 ( c cdot 4 ) D. 5 | 11 |

1680 | Which term of the sequence 72,70,68 ( 66, dots ) is ( 40 ? ) A . 14 B. 15 ( c cdot 16 ) ( D cdot 17 ) | 11 |

1681 | Find the Odd one among : 7, 26, 65, 124, 215,342 A. 7 B. 26 ( c cdot 65 ) D. 124 | 11 |

1682 | The sum of odd integers from 1 to 2001 is A. ( 1001^{2} ) B. ( 1000^{2} ) ( mathbf{c} cdot 1002^{2} ) D. ( 1003^{2} ) | 11 |

1683 | In the following ( boldsymbol{A P} ), find the missing terms in the boxes: ( square, 13, square, 3 ) | 11 |

1684 | If ( a, b, c ) are in A.P., then ( frac{1}{sqrt{b}+sqrt{c}} ) ( frac{1}{sqrt{boldsymbol{c}}+sqrt{boldsymbol{a}}}, frac{1}{sqrt{boldsymbol{a}}+sqrt{boldsymbol{b}}} ) are in A . G.P. B. H.P. c. A.P. D. None of these | 11 |

1685 | If 7 times the ( 7^{t h} ) term of an ( A . P geq 11 ) times its ( 11^{t h} ) term, then find its ( 18^{t h} ) term. | 11 |

1686 | Find the last term of the following sequence ( 2,4,8 dots dots dots ) to 9 terms | 11 |

1687 | 15th term of the series ( 243,81,27, ldots ) is This question has multiple correct options ( ^{mathbf{A}} cdotleft(frac{1}{3}right)^{9} ) ( ^{text {B }}left(frac{1}{3}right)^{10} ) ( ^{text {c. }}left(frac{1}{3}right)^{10}left(frac{1}{3}right)^{-1} ) ( ^{mathrm{D} cdot}left(frac{1}{3}right)^{10}left(frac{1}{3}right) ) | 11 |

1688 | Find the sum of the first 16 terms of the A.P.10,6,2,.. | 11 |

1689 | Write an A.P. having 4 as the first term and -3 common difference. | 11 |

1690 | Identify whether the following sequence is a geometric sequence or not ( mathbf{1}, mathbf{4}, mathbf{9}, mathbf{1 6} ) | 11 |

1691 | Find the value of ( n, ) if ( 1+4+7+10+ ) ( ldots ) to ( n ) terms ( =mathbf{5 9 0} ) | 11 |

1692 | Which one of the following is not a series? A. adding first ( n ) natural numbers B. multiplying first 10 odd numbers c. adding first 20 even numbers D. adding last 20 natural numbers | 11 |

1693 | Let ( x_{1}, x_{2}, ldots . . ) be positive integers in A.P., such that ( x_{1}+x_{2}+x_{3}=12 ) and ( boldsymbol{x}_{4}+boldsymbol{x}_{6}=mathbf{1 4} . ) Then ( boldsymbol{x}_{5} ) is This question has multiple correct options A. A prime number B. 11 ( c cdot 13 ) D. 7 | 11 |

1694 | ( sum_{p=1}^{32}(3 p+2)left[sum_{q=1}^{10}left(sin frac{2 q pi}{11}-i cos frac{2 q pi}{11}right)right. ) A ( cdot 8(1-i) ) B ( cdot 16(1-i) ) ( mathbf{c} cdot 48(1-i) ) D. None of these | 11 |

1695 | ( mathbf{1} . mathbf{3}+mathbf{3 . 5}+mathbf{5 . 7}+ldots+(mathbf{2 n}-mathbf{1})(mathbf{2 n}+ ) 1) ( = ) A ( frac{nleft(4 n^{2}+6 n-1right)}{3} ) в. ( frac{nleft(3 n^{2}+5 n+1right)}{3} ) ( frac{nleft(5 n^{2}+7 n-1right)}{3} ) D. ( frac{nleft(7 n^{2}-5 n+1right)}{3} ) | 11 |

1696 | The minimum value of the sum of real numbers ( a^{-5}, a^{-4}, 3 a^{-3}, 1, a^{8} ) and ( a^{10} ) with ( a>0 ) is ( mathbf{A} cdot mathbf{6} ) B. 7 c. 8 D. | 11 |

1697 | (u) IUT 31. If, for a positive integer n, the quadratic equation, x(x + 1) + (x + 1)(x + 2) + … + (x+ n-1) (x +n) =10 has two consecutive integral solutions, then n is equal (JEEM 2017] (a 11 (b) 12 (C) 9 (d) 10 | 11 |

1698 | A divergent series: A. The infinite sequence of the partial sums of the series does not have a finite limit. B. ( 2+4+6+8+ldots . . ) C. Both A and B are correct D. Only A is correct | 11 |

1699 | Let ( a_{n} ) be an A.P. for which ( d=8 ) and ( a_{2}=12 . ) Find ( a_{1} ) A . 1 B. 4 ( c .3 ) D. 2 | 11 |

1700 | If the geometric mean of three observations 40,50 and ( x ) is ( 10, ) then the value of ( x ) is A ( cdot frac{1}{2} ) B. 4 ( c .6 ) D. | 11 |

1701 | A sequence in which the difference between any two consecutive terms is a constant is called as A. G.P B. A.P. c. н.P D. A.G.P | 11 |

1702 | Find the number of terms in the series ( 20+19 frac{1}{3}+18 frac{2}{3}+ldots . ) of which the sum is 300 , explain the double answer. | 11 |

1703 | Find the sum of the series: ( 1^{2}-2^{2}+ ) ( 3^{2}-4^{2}+ldots ) to ( n ) terms. | 11 |

1704 | Find the ( 5^{t h} ) term of the ( A . P . ) ( 17,14,11, dots dots,-40 ) | 11 |

1705 | ( mathbf{1}+mathbf{3}+mathbf{6}+mathbf{1 0}+ldots+frac{(boldsymbol{n}-mathbf{1}) boldsymbol{n}}{mathbf{2}}+ ) ( frac{boldsymbol{n}(boldsymbol{n}+mathbf{1})}{mathbf{2}}= ) A. ( frac{n(n+1)(n+2)}{3} ) B. ( frac{(n+1)(n+2)}{6} ) c. ( frac{n(n+1)(n+2)}{6} ) D. ( frac{(n+2)(n+1)^{2}}{3} ) | 11 |

1706 | What is value of ( 1+x+x^{2}+x^{3}+x^{4}+dots ) where ( boldsymbol{x} neq mathbf{1} ) | 11 |

1707 | Let ( A, G ) and ( H ) be the ( A M, G M ) and ( H M ) of two positive numbers a and b. The quadratic equation whose roots are ( boldsymbol{A} ) and ( H ) is This question has multiple correct options ( mathbf{A} cdot A x^{2}-left(A^{2}+G^{2}right) x+A G^{2}=0 ) B ( cdot A x^{2}-left(A^{2}+H^{2}right) x+A H^{2}=0 ) ( mathbf{C} cdot H x^{2}-left(H^{2}+G^{2}right) x+H G^{2}=0 ) D. ( G x^{2}-left(H^{2}+G^{2}right) x+G H^{2}=0 ) | 11 |

1708 | In a geometric progression, the sum of first ( n ) terms is ( 65535 . ) If the last term is 49152 and the common ratio is ( 4, ) then find the value of ( n ) | 11 |

1709 | Find all possible integers whose geometric mean is 16 A ( .(1,256),(2,128),(4,64),(8,32),(16,16) ) B . (1,256),(2,128),(4,64),(16,1),(1,16) C ( .(1,256),(4,4),(4,64),(8,32),(16,16) ) D. (1,256),(2,128),(1,4),(8,32),(16,16) | 11 |

1710 | If the ( p^{t h} ) term of the series of positive numbers ( 25,22 frac{3}{5}, 20 frac{1}{2}, 18 frac{1}{4}, dots ) is numerically the smallest, then the ( p^{t h} ) is. A ( cdot frac{1}{4} ) B. c. ( frac{1}{3} ) D. | 11 |

1711 | The sum of four numbers in AP is 176 The product of 1st and last is 1855. The mean of middle two is ( mathbf{A} cdot 42 ) B. 41 c. 44 D. 53 | 11 |

1712 | Find the smallest positive integer ( n ) such that ( t_{n} ) of the arithmetic sequence ( 20,19 frac{1}{4}, 18 frac{1}{2}, ldots . ) is negative? | 11 |

1713 | The arithmetic mean of 5,6,8,9,12,13 17 is A . 20 B. 15 c. 10 D. 25 | 11 |

1714 | Find the value of ( x ) for which ( (5 x+2),(4 x-1) ) and ( (x+2) ) are in A.P | 11 |

1715 | If ( frac{1}{p+q}, frac{1}{r+p}, frac{1}{q+r} ) are in A.P., then A ( cdot p^{2}, q^{2}, r^{2} ) are in A.P. B . ( q^{2}, p^{2}, r^{2} ) bare in A.P. c. ( q^{2}, r^{2}, p^{2} ) are in A.P. D. ( p, q, r ) are in A.P. | 11 |

1716 | sum of the series is ( 1+3+6+10+ ) ( 15+ldots ) to ( n ) terms ( frac{n(n+m)(n+p)}{k} ) Find ( boldsymbol{k}-boldsymbol{m}-boldsymbol{p} ? ) | 11 |

1717 | 27. u 25 Ifa,,a….,,, are positive real numbers whose product is a fixed number c, then the minimum value of a, +a, + …….+a-1 + 2a, is (2002) (a) n(2c)’n () (n+1) c/ (c) 2ncin (d) (n+1)(2c) ‘n | 11 |

1718 | Find the sum of the products of the corresponding terms of the following sequences:- (i) 2,4,8,16,32 (ii) ( 128,32,8,2, frac{1}{2} ) | 11 |

1719 | Is the given Progression arithmetic progression?Why ( 2,6,7,10,12,15, dots dots dots ) | 11 |

1720 | ( s_{n}=5 n^{2}+11 n ) Find ( s_{5} ) | 11 |

1721 | f the ( m ) th term of an AP is ( a ) and its ( n ) th term is ( b ), show that the sum of its ( (m+n) ) terms is ( frac{(m+n)}{2}left{a+b+frac{(a-b)}{(m-n)}right} ) | 11 |

1722 | If ( frac{boldsymbol{a}-boldsymbol{x}}{boldsymbol{p} boldsymbol{x}}=frac{boldsymbol{a}-boldsymbol{y}}{boldsymbol{q} boldsymbol{y}}=frac{boldsymbol{a}-boldsymbol{z}}{boldsymbol{r} boldsymbol{z}} ) and ( boldsymbol{p}, boldsymbol{q}, boldsymbol{r} ) be in A.P. then ( x, y, z ) are in? A. A.P в. G.P. c. н.P D. A.G.P. | 11 |

1723 | Sum to 20 terms of the series ( 1.3^{2}+ ) ( 2.5^{2}+3.7^{2}+ldots ) is A . 178090 B. 168090 c. 188090 D. None of these | 11 |

1724 | If the ( 3^{r d} ) and the ( 9^{t h} ) terms of an ( A P ) are 4 and -8 respectively, then which term of this ( A P ) is zero | 11 |

1725 | Identify the finite geometric progression. A ( .3,6,12,24 . ) в. ( 81,27,9,3 . ) c. ( 10-5+2.5-1.25 dots ) D. ( 1+0.5+0.25+0.125 ) | 11 |

1726 | Solve: ( frac{1}{2}+frac{1}{6}+frac{1}{12}+frac{1}{20}+ldots . .+ ) ( frac{1}{1892}+frac{1}{1980}=? ) | 11 |

1727 | Find the ( 7^{t h} ) term of the G.P. ( 2,-6,18, dots ) A. 1458 B. 2458 8 ( c .3458 ) D. 4458 | 11 |

1728 | If the ( m^{t h} ) term of an A.P be ( frac{1}{n}, n^{t h} ) term be ( frac{1}{m}, ) show that it’s ( m n^{t h} ) term is 1 | 11 |

1729 | Maximum value of ( prodleft(1+frac{b-c}{a}right)^{a} ) is ( lambda ) where ( a, b, c ) are integral sides of a triangle, ( lambda+2 ) is/are divisible by This question has multiple correct options A . 2 B. 3 c. 1 D. 5 | 11 |

1730 | The ( 5^{t h} ) and ( 8^{t h} ) terms of a GP are 1458 and ( 54, ) respectively. The common ratio of the GP is A ( cdot frac{1}{3} ) B. 3 ( c cdot 9 ) D. ( frac{1}{9} ) E | 11 |

1731 | If roots of the equations ( (b-c) x^{2}+ ) ( (c-a) x+a-b=0, ) where ( b neq c, ) are equal, then a, b, c are in? A. G.P B. H.P. ( c cdot ) A.P D. A.G.P. | 11 |

1732 | If the ratio of sum of ( n ) terms in two A.P’s is ( 2 n: n+1, ) then the ratio of ( 8^{t h} ) terms is ( mathbf{A} cdot 15: 8 ) B. 8: 133 ( mathbf{c} .5: 17 ) D. none | 11 |

1733 | ( (x-4) ) is geometric mean of ( (x-5) ) and ( (x-2) ) find ( x ) | 11 |

1734 | If ( frac{2+5+8+ldots n text { terms }}{7+11+15+ldots . n text { terms }}=frac{23}{35} ) then ( n ) value is A . 17 B . 15 c. 18 D. 23 | 11 |

1735 | ( p^{t h} ) term of the series ( left(3-frac{1}{n}right)+left(3-frac{2}{n}right)+left(3-frac{3}{n}right) dots dots ) will be ( A cdot 3+frac{p}{n} ) B ( cdot 3-frac{p}{n} ) c. ( 3+frac{n}{p} ) D. ( 3-frac{n}{p} ) | 11 |

1736 | In a G.P. of positive terms, if any term is equal to the sum of next two terms, find the common ratio of the G.P. | 11 |

1737 | Which term of the geometric sequence, (i) ( 5,2, frac{4}{5}, frac{8}{25}, dots, ) is ( frac{128}{15625} ? ) (ii) ( 1,2,4,8, dots, ) is ( 1024 ? ) | 11 |

1738 | The G.M. of first n natural numbers is A ( cdot frac{n+1}{2} ) в. ( (n !)^{n} ) ( mathbf{c} cdot(n !)^{1 / n} ) D. None of these | 11 |

1739 | If 2 nd, 3 rd and 6 th terms of an AP are the three consecutive terms of a GP then find the common ratio of the GP. | 11 |

1740 | f ( a, b, c ) are in G.P., then A ( cdot aleft(b^{2}+a^{2}right)=cleft(b^{2}+c^{2}right) ) B . ( aleft(a^{2}+c^{2}right)=cleft(a^{2}+b^{2}right) ) C ( cdot a^{2}(b+c)=c^{2}(a+b) ) D. None of these | 11 |

1741 | Solve ( 1.2+2.2^{2}+3.2^{3}+ldots+n .2^{n}=(n- ) 1) ( 2^{n+1}+2 ) | 11 |

1742 | Which of the following are APs? This question has multiple correct options ( mathbf{A} cdot 2.4,8,16, ldots ) B. ( 2, frac{5}{2}, 3, frac{7}{2}, ldots ) C ( .-1.2,-3.2,-5.2,-7.2, ldots ) D. ( -10,-6,-2,2, ldots ) E ( .3,3+sqrt{2}, 3+2 sqrt{2}, 3+3 sqrt{2}, ldots ) F. ( 0.2,0.22,0.222,0.2222, ldots . ) G. ( 0,-4,-8,-12, dots ) ( mathrm{H} cdot-frac{1}{2},-frac{1}{2},-frac{1}{2},-frac{1}{2}, ldots ) | 11 |

1743 | ( 4+8+16+32+ldots . . ) What is the common ratio? ( mathbf{A} cdot mathbf{1} ) B . 2 ( c .3 ) D. 4 | 11 |

1744 | Calculate the sum of the series ( left(x+frac{1}{x}right)^{2}+left(x^{2}+frac{1}{x^{2}}right)^{2}+ ) ( left(boldsymbol{x}^{3}+frac{1}{boldsymbol{x}^{3}}right)^{2}+ldots . .left(boldsymbol{x}^{boldsymbol{n}}+frac{1}{boldsymbol{x}^{boldsymbol{n}}}right)^{2} ) A ( cdot frac{x^{n}-1}{x^{2}-1} cdotleft[frac{x^{2 n+2}+1}{x^{2 n}}right]+2 n ) B. ( frac{x^{n}-1}{x^{2}-1} cdotleft[frac{x^{n+1}+1}{x^{2 n}}right]+2 n ) ( frac{x^{2 n}-1}{x^{2}-1} cdotleft[frac{x^{2 n+2}+1}{x^{2 n}}right]+2 n ) D ( frac{x^{2 n}-1}{x^{2}-1} cdotleft[frac{x^{n+1}+1}{x^{2 n}}right]+2 n ) | 11 |

1745 | The first term of an ( A . P ) is ( p ) and its common difference is ( q ). Find its 10 th term | 11 |

1746 | If ( boldsymbol{x}=frac{mathbf{1}}{mathbf{1}^{2}}+frac{mathbf{1}}{mathbf{3}^{2}}+frac{mathbf{1}}{mathbf{5}^{2}}+ldots ., boldsymbol{y}=frac{mathbf{1}}{mathbf{1}^{2}}+ ) ( frac{3}{2^{2}}+frac{1}{3^{2}}+frac{3}{4^{2}}+ldots ) and ( z=frac{1}{1^{2}}- ) ( frac{1}{2^{2}}+frac{1}{3^{2}}-frac{1}{4^{2}}+ldots ., ) then A. ( x, y, z ) are in A.P. B. ( frac{y}{6}, frac{x}{3}, frac{z}{1} ) are in A.P. c. ( frac{y}{6}, frac{x}{3}, frac{z}{2} ) are in A.P D. ( 6 y, 3 x, 2 z ) are in H.P | 11 |

1747 | The second, the first, and the third term of an arithmetic progression, whose common difference is nonzero, form a geometric progression in that order. Find its common ratio. | 11 |

1748 | If the natural numbers are divided into groups of {1},{2,3},{4,5,6},{7,8,9,10} … Then the /sum of 50th group is A. 65225 B. 56225 c. 62525 D. 53625 | 11 |

1749 | Find the ( A M, G M ) and HM between 9 and 18 | 11 |

1750 | The fifth and fifteenth term of an ( A . P . ) are 13 and ( -17 . ) find sum of first 21 terms of an ( boldsymbol{A} boldsymbol{. P .} ) | 11 |

1751 | In a triangle ( A B C, ) bisector of angle ( C ) meets the side ( A B ) at ( D ) and circumcentre at E. The maximum value of CD.DE is equal to A ( cdot a^{2} / 4 ) B . ( b^{2} / 4 ) c. ( c^{2} / 4 ) D. ( (a+b)^{2} / 4 ) | 11 |

1752 | Let ( a_{n} ) be the nth term of the G.P. of positive numbers. Let ( sum_{n=1}^{100} a_{2 n}=alpha ) and ( sum_{n=1}^{100} a_{2 n-1}=beta, ) such that ( alpha neq beta . ) Prove that the common ratio of the G.P. is ( frac{boldsymbol{alpha}}{boldsymbol{beta}} ) | 11 |

1753 | The harmonic mean of two numbers is 4. The arithmetic mean ( A ) & the geometric mean G satisfy the relation ( mathbf{2} boldsymbol{A}+boldsymbol{G}^{2}=mathbf{2} 7 . ) Find the largest of those numbers? ( mathbf{A} cdot mathbf{6} ) B. 4 c. 8 D. 5 | 11 |

1754 | Tha angles of a quadrilateral are in A.P., whose common difference is ( 10^{circ} ). Find the angles. | 11 |

1755 | Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively. | 11 |

1756 | For the infinite series ( 1-frac{1}{2}-frac{1}{4}+frac{1}{8}- ) ( frac{1}{16}-frac{1}{32}+frac{1}{54}-frac{1}{128}-ldots . quad ) let ( S ) be the (limiting) sum. Then ( S ) equals A. 0 в. ( frac{2}{7} ) ( c cdot frac{6}{7} ) D. ( frac{9}{32} ) E ( frac{27}{32} ) | 11 |

1757 | A geometric series consists of four terms and has a positive common ratio. The sum of the first two terms is 9 and sum of the last two terms is ( 36 . ) Find the series | 11 |

1758 | The sequence ( -6+42-294+2058 ) is ( a ) A . finite geometric sequence B. finite arithmetic sequence C. infinite geometric sequence D. infinite harmonic sequence | 11 |

1759 | The difference of the squares of two consecutive even integers is divisible by which of the following integers? ( A cdot 3 ) B. 4 ( c .6 ) D. 7 | 11 |

1760 | In any G.P. the first term is 2 and last term is 512 and common ratio is ( 2, ) then ( 5^{t h} ) term from end is – A . 16 B. 32 ( c cdot 64 ) D. None of these | 11 |

1761 | Write the formula for the sum of first ( n ) positive integers. | 11 |

1762 | Given the following sequence, determine whether it is arithmetic progression or not. If it is an Arithmetic Progression, its general term. ( -5,2,9,16,23,30, dots dots dots ) A ( .7 n-12 ) в. ( 6 n-12 ) ( c cdot 5 n-12 ) D. ( 4 n-12 ) | 11 |

1763 | Find ( a_{1}+a_{6}+a_{11}+a_{16} ) if it is known that ( a_{1}, a_{2} dots . ) is an A.P. and ( a_{1}+a_{4}+ ) ( boldsymbol{a}_{boldsymbol{7}}+ldots+boldsymbol{a}_{mathbf{1 6}}=mathbf{1 4 7} ) | 11 |

1764 | 39. Let the sum of the first n terms of a non-constant A.P., a, a n(n-7) – A, where A is a constant az, ………….. be 50n + If d is the common difference of this A.P., then the ordered pair (d, aço) is equal to: JEEM 2019-9 April (M) (a) (50,50+46A) (b) (50, 50+45A) (C) (A, 50+45A) (d) (A, 50 +46A) | 11 |

1765 | 22. Let S be a square of unit area. Consider any quadrilateral which has one vertex on each side of S. If a, b, c, and d denote the lengths of the sides of the quadrilateral, prove that 2 sa2+ba+c2+d- 54. (1997-5 Marks) | 11 |

1766 | Prove that ( frac{(2 n !)}{n !}=2^{n}(1.3 .5 dots .(2 n-1)) ) | 11 |

1767 | 26. Three positive numbers form an increasing GP term in this G.P. is doubled, the new numbers are in A. the common ratio of the G.P. is: (a) 2- 3 (b) 2+3 (c) 5.5 (d) 3+12 rm an increasing G. P. If the middle he new numbers are in A.P. then [JEEM 2014] | 11 |

1768 | ( ln ) a ( G . P ) sum of ( n ) terms is 255 , the last term is 128 and the common ratio is 2 . Find ( n ) | 11 |

1769 | Which of the following is in the form of A.P.? в. ( 0,3,2,1,-2 ldots ) c. 4,5,7,10,14 D. ( -2,2,-2,2,-2 dots ) | 11 |

1770 | Write any two arithmetic progressions with common difference 4 | 11 |

1771 | 1. The equation 2 cosa-sin? x = x2 + x-2; 0<x< has 2 a. no real solution b. one real solution c. more than one solution d. none of these (IIT-JEE 1980) | 11 |

1772 | The sum ( frac{1}{1+1^{2}+1^{4}}+frac{2}{1+2^{2}+2^{4}}+ ) ( frac{mathbf{3}}{mathbf{1}+mathbf{3}^{2}+mathbf{3}^{4}}+ldots+frac{mathbf{9 9}}{mathbf{1}+mathbf{9 9}^{2}+mathbf{9 9}^{4}} ) lies between A. 0.46 and 0.47 7 B. 0.52 and 1.0 c. 0.48 and 0.49 D. 0.49 and 0.50 | 11 |

1773 | The sum of the first five terms an AP and the sum of the first seven terms of the same ( A P ) is 167 . If the sum of the first ten terms of this ( A P ) is 235 , find the sum of its first twenty terms. A . -4230 в. 4230 c. -2430 D. -3240 | 11 |

1774 | ( ln ) an A.P. the first term is 2 and the sum of the first five terms is one fourth of the next five terms. Show that 20 th term is -112 | 11 |

1775 | Find the tenth term of G.P : ( mathbf{5}, mathbf{2 5}, mathbf{1 2 5}, dots dots ) | 11 |

1776 | 16. If a1, a…….., an are in H.P., then the expression [2006] aya2 + azaz + ………. + an-1an is equal to (a) n(Q1 – an) (b) (n-1)(aj -an). (c) najan (d) (n-1)ajan | 11 |

1777 | For the following arithmetic progressions write the first term ( a ) and the common difference ( d: ) ( frac{1}{5}, frac{3}{5}, frac{5}{5}, frac{7}{5}, dots . . ) | 11 |

1778 | If the A.M. of the roots of a quadratic equation is ( frac{8}{5} ) and ( A . M . ) of their reciprocals is ( frac{8}{7}, ) then the quadratic equation is A ( cdot 5 x^{2}-8 x+7=0 ) B. ( 5 x^{2}-16 x+7=0 ) c. ( 7 x^{2}-16 x+5=0 ) D. ( 7 x^{2}+16 x+5=0 ) | 11 |

1779 | How many terms of the geometric progression ( 1+4+16+64+dots dots ) must be added to get sum equal to ( mathbf{5 6 4 1} ? ) | 11 |

1780 | State True or False. f ( x, y ) are positive real numbers such that ( x+y=1, ) then ( left(1+frac{1}{x}right)left(1+frac{1}{y}right) geqslant 9 ) A. True B. False | 11 |

1781 | An AP consists of 50 terms of which ( 3^{r d} ) term is 12 and the last term is ( 106 . ) Find the ( 29^{t h} ) term. | 11 |

1782 | Find the sum the infinite G.P.: ( 1+frac{1}{3}+frac{1}{9}+frac{1}{27}+dots dots ) A ( cdot frac{3}{5} ) B. ( frac{3}{2} ) c. ( frac{49}{27} ) D. 8 | 11 |

1783 | The sum of ( n ) terms of three arithmetical progression are ( S_{1}, S_{2} ) and ( S_{3} . ) The first term of each is unity and the common differences are 1,2 and 3 respectively. Prove that ( left(boldsymbol{S}_{1}+boldsymbol{S}_{3}=boldsymbol{2} boldsymbol{S}_{2}right) ) | 11 |

1784 | Let ( A ) be the sum of the first 20 terms and ( B ) be the sum of the first 40 terms of the series ( 1^{2}+2.2^{2}+3^{2}+2.4^{2}+ ) ( 5^{2}+2.6^{2}+ldots . . ) If ( B-2 A=100 lambda, ) then ( lambda ) is equal to ( mathbf{A} cdot 464 ) в. 496 c. 232 D. 248 | 11 |

1785 | Find ( A M ) of 12 and 14 | 11 |

1786 | If the sum to ( n ) terms of an AP is ( frac{4 n^{2}-3 n}{4} ) then the ( n^{t h} ) term of the AP is equal to A ( cdot frac{5 n-1}{4} ) в. ( frac{8 n-7}{4} ) c. ( frac{3 n^{2}-2}{4} ) D. ( frac{7 n-8}{4} ) | 11 |

1787 | If ( a+b+c+d+e+f=12 ) then the maximum value of ( a b+b c+c d+d e+ ) ( e f+f a ) is ( (a, b, c, d, e, f ) are non negative real numbers) A . 36 B . 24 c. 30 D. none of these | 11 |

1788 | The A.M. of ( 1,3,5, dots,(2 n-1) ) is- ( mathbf{A} cdot n+1 ) B. ( n+2 ) ( c cdot n^{2} ) D. | 11 |

1789 | The A.M. of two numbers is 34 and their G.M. is ( 16 . ) The two numbers are A . 60,8 в. 64,4 c. 56,12 D. 52,16 | 11 |

1790 | The sum ( frac{3}{1.2}, frac{3}{1.2}, frac{1}{2}, frac{4}{2.3}left(frac{1}{2}right)^{2}+frac{5}{3.4}left(frac{1}{2}right)^{2} ) A ( cdot 1-frac{1}{(n+1) 2^{n}} ) B. ( 1-frac{1}{n .2^{n-1}} ) ( mathbf{C} cdot 1=frac{1}{(n+1) 2^{n}} ) D. ( frac{1}{(n-1) 2^{n-1}} ) | 11 |

1791 | Three distinct numbers, ( x, y, z ) form a geometric progression in that order, and the numbers ( boldsymbol{x}+boldsymbol{y}, boldsymbol{y}+boldsymbol{z}, boldsymbol{z}+boldsymbol{x} ) form an arithmetic progression in that order. Find the common ratio of the geometric progression. | 11 |

1792 | What is the geometric mean of the sequence ( 1,2,4,8, dots dots 2^{n} ? ) ( mathbf{A} cdot 2^{n / 2} ) B. ( 2^{(n+1) / 2} ) c. ( 2^{(n+1)}-1 ) D. ( 2^{(n-1)} ) | 11 |

1793 | If the product of the first four consecutive terms of a G.P is 256 and if the common ratio is 4 and the first term is positive, then its ( 3^{r d} ) term is ( A cdot 8 ) B. ( frac{1}{16} ) c. ( frac{1}{32} ) D. 16 | 11 |

1794 | The maximum value of ( mathbf{f}(mathbf{x})= ) ( frac{1}{2 e^{x}+e^{-x}} ) is: A ( cdot frac{1}{2 sqrt{2}} ) B. ( 2 sqrt{2} ) ( c cdot frac{1}{2} ) D. ( frac{mathrm{e}}{2 mathrm{e}^{2}+1} ) | 11 |

1795 | If the first, second and last terms of an A.P be ( a, b, 2 a ) respectively, then its sum will be A ( cdot frac{a b}{-a+b} ) В. ( frac{a b}{2(b-a)} ) c. ( frac{3 a b}{2(b-a)} ) D. ( frac{3 a b}{4(b-a)} ) | 11 |

1796 | 5th term of an AP is 26 and 10 th term is 51. The 15 th term is : A . 60 B. 76 ( c .55 ) D. 72 | 11 |

1797 | ( f(1+x)left(1+x^{2}right)left(1+x^{4}right) dots .(1+ ) ( left.x^{128}right)=sum_{r=0}^{n} x^{r} ) then findthe value of ( n ) | 11 |

1798 | Find the 1000 th term of the sequence ( 3,4,5,6, dots ) | 11 |

1799 | Between two numbers whose sum is ( frac{13}{6} ) an even number of A.M.s are inserted, the sum of these means exceeds their number by unity. Find the number of means. | 11 |

1800 | if ( frac{1}{1^{2}}+frac{1}{2^{2}}+frac{1}{3^{2}}+ldots . . . . . ) upto ( infty=frac{pi^{2}}{6} ) ( operatorname{then} frac{1}{1^{2}}+frac{1}{3^{2}}+frac{1}{5^{2}}+ldots ) A ( cdot pi^{2} / 8 ) в. ( pi^{2} / 12 ) c. ( pi^{2} / 3 ) D . ( pi^{2} / 9 ) | 11 |

1801 | Find the general term ( left(n^{t h} ) term ) and right. ( 23^{t h} ) term of the sequence ( mathbf{3}, mathbf{1},-mathbf{1},-mathbf{3}, dots ) | 11 |

1802 | If there exists a geometric progression containing 27,8 and 12 as three of its terms (not necessarily consecutive) then no. of progressions possible are A . 1 B . 2 c. infinite D. None of these | 11 |

1803 | Calculate the 15 th term of the A.P. ( -3,-4,-5,-6,-7 dots ) A . -13 в. -15 ( c .-17 ) D. -19 | 11 |

1804 | For all real numbers ( a, b ) and positive integer ( n ) prove that: ( (a+b)^{n}=^{n} C_{0} a^{n}+^{n} C_{1} a^{n-1} b+^{n} ) ( C_{1} a^{n-2} b^{2}+ldots ldots ldots .+^{n} C_{n} b^{n} ) | 11 |

1805 | The sum of the first three terms of an increasing geometric progression is 13 and their product is ( 27 . ) Calculate the sum of the first five terms of the progression. | 11 |

1806 | ( ln operatorname{an} A P a=5, d=3, a_{n}=50 ) find ( n ) and ( boldsymbol{S}_{n} ) | 11 |

1807 | The common ratio is used in progression. A . arithmetic B. geometric c. harmonic D. series | 11 |

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