Sequences And Series Questions

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} )
( mathbf{B} cdot frac{n(n+1)left(3 n^{2}+7 n+8right)}{12} )
( mathbf{c} cdot frac{n(n+1)(n+2)left(n^{2}+5 n+6right)}{12} )
( mathbf{D} cdot frac{(mathbf{n}+1)(mathbf{n}+2)(mathbf{n}+3)}{4} )

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
A. ( n(n+1) )
B ( cdot n^{2} )
c. ( n(4-n) )
D. ( n(6-n) )

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
common ratio ( r, ) then prove that ( frac{1}{a_{1} m+a_{2} m}+frac{1}{a_{2} m+a_{3} m}+dots+ )
( frac{1}{a_{n-1}^{m}+a_{n-1}^{m}}=frac{1-r^{m(1-n)}}{a_{1}^{m}left(r^{m}-r^{-m}right)} )

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
show that the ( 29^{t h} ) term is twice the
( 19^{t h} ) term
A . True
B. False

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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