# Principle Of Mathematical Induction Questions

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

Question NoQuestionsClass
1Prove that ( 1+2+3+ldots .+n= )
( frac{n(n+1)}{2} . ) for ( n ) being a natural
numbers.
11
2The inequality ( n !>2^{n-1} ) is true
( mathbf{A} cdot ) for all ( n>1 )
B. for all ( n>2 )
c. for all ( n epsilon N )
D. none of these
11
3Prove for ( boldsymbol{n} in mathbb{N} )
( mathbf{1} . mathbf{3}+mathbf{2} . mathbf{3}^{2}+mathbf{3} . mathbf{3}^{3}+ldots+boldsymbol{n} . mathbf{3}^{n}= )
( frac{(2 n-1) 3^{n+1}+3}{4} )
11
4For each ( n epsilon N, ) then ( 3^{2 n+1}+1 ) is
divisible by –
( A cdot 2 )
B. 3
( c cdot 7 )
D. None of these
11
5Prove the following by using principle of mathematical induction for all ( boldsymbol{n} in boldsymbol{N} )
( mathbf{1} . mathbf{3}+mathbf{3 . 5}+mathbf{5 . 7}+ldots ldots .+(mathbf{2 n}- )
( mathbf{1})(mathbf{2 n}+mathbf{1})=frac{boldsymbol{n}left(mathbf{4} boldsymbol{n}^{2}+mathbf{6} boldsymbol{n}-mathbf{1}right)}{mathbf{3}} )
11
6Let p be a prime and m a positive integer. By mathematical
induction on m, or otherwise, prove that whenever r is an
integer such that p does not divide r, p divides mp ,
(1998 -8 Marks)
(Hint: You mayuse the fact that (1+x)(m+1)=(1+x)(1+x)mpil
11
7Using the principle of mathematical induction, show that;
( mathbf{2}+mathbf{4}+mathbf{6}+ldots .+mathbf{2} boldsymbol{n}=boldsymbol{n}^{2}+boldsymbol{n} )
11
827.
Use mathematical induction to show that
(25)n+1-24n+ 5735 is divisible by (24)2 for all n=1,2,……
(2002 – 5 Mark
(20032 Mart
11
9Mathematical Induction is the principle
containing the set
( A cdot R )
B.
( c cdot Q )
( D )
11
10If ( boldsymbol{P}(boldsymbol{n})=mathbf{1}^{2}+mathbf{3}^{2}+mathbf{5}^{2}+ldots+ )
( (2 n-1)^{2}=frac{nleft(4 n^{2}-1right)}{3}, ) then which of
the following does NOT hold good?
( mathbf{A} cdot P(1) )
B ( cdot P(2) )
( mathbf{c} cdot P(3) )
D. None of these.
11
11Using the principle of mathematical induction, find ( tan alpha+2 tan 2 alpha+ )
( 2^{2} tan 2^{2} alpha+ldots . ) to ( n ) terms:
A ( cdot tan alpha-2^{n} cdot tan left(2^{n} cdot alpharight) )
B. ( cot alpha-2^{n} cdot cot left(2^{n} cdot alpharight) )
c. ( sec alpha-2^{n} cdot sec left(2^{n} cdot alpharight) )
D. None of these
11
122.
Prove that 72n † (23n – 3)(3n – 1) is divisible by 25 for any
natural number n.
(1982 – 5 Marks).
11
13If ( a_{1}=1, a_{n+1}=frac{1}{n+1} a_{n}, forall n geq 1, ) then
( boldsymbol{a}_{boldsymbol{n}}= )
A ( cdot frac{1}{n !} )
в. ( frac{1}{(n+2) !} )
c. ( frac{1}{(n+1) !} )
D. none of these
11
14By the principle of Mathematical induction, prove that, for ( n geq 1 )
[
mathbf{1}^{2}+mathbf{2}^{2}+mathbf{3}^{2}+cdots+boldsymbol{n}^{2}>frac{boldsymbol{n}^{mathbf{3}}}{mathbf{3}}
]
11
15For all ( n in N, n^{4} ) is less than
A ( cdot 10^{n} )
B . ( 4^{n} )
( c cdot 10^{10} )
D. None of these
11
16Prove that ( 25^{n}-20^{n}-8^{n}+3^{n} ) is
divisible by 5
11
17( forall boldsymbol{n} in boldsymbol{N}, boldsymbol{2} cdot boldsymbol{4}^{2 n+1}+boldsymbol{3}^{3 n+1} ) is divisible by
( A cdot 7 )
B. 5
c. 11
D. 209
11
18(1983 – 2 Marks)
Ifp be a natural number then prove that pn+1 + (p + 1)2n-1
is divisible by p2 + p + 1 for every positive integer n.
(1984 – 4 Marks)
11
19Prove that induction that
( sin x+sin 3 x+dots+sin (2 n-1) x= )
( frac{sin ^{2} n x}{sin x} )
11
20Applying the principal of mathematical
induction of prove that
[
frac{2 sin theta}{cos theta+3 cos theta}+frac{2 sin theta}{cos theta+5 cos theta}+
]
( frac{2 sin theta}{cos theta+cos (2 n+1) theta} )
[
=tan (n+1) theta-tan theta
]
11
21Prove the following by using the principle of mathematical induction for
all ( boldsymbol{n} in boldsymbol{N} )
( mathbf{1}+mathbf{3}+mathbf{3}^{2}+ldots+mathbf{3}^{n-1}=frac{left(mathbf{3}^{n}-mathbf{1}right)}{mathbf{2}} )
11
22– TUPILWIDU DO
13.
Using induction or otherwise, prove that for any non
negative integers m, n, r and k,
(1991 – 4 Marks)
(n-m
(r + m)!

(r + k +1)!
k!
n
r +1
m!
m=0
r +2
11
23Find counter examples to disprove the following statement

A quadrilateral with all sides are equal
is a square.

11
24Using Principle of mathematical
induction prove that ( 6^{n}-1 ) divisible by
( mathbf{5} )
11
25Let ( P(n): n^{2}+n ) is an odd integer. It is
seen that truth of ( P(n) Rightarrow ) the truth of ( P(n+1) . ) Therefore, ( P(n) ) is true for all
( A cdot n>1 )
B.
( c cdot n>2 )
D. none of these
11
26( 7^{2 n}+2^{3 n-3} .3^{n-1} ) is divisible by 25 for
any natural number ( n geq 1 . ) Prove that by
mathematical
11
27Use the principle of mathematical induction to prove that ( boldsymbol{a}+(boldsymbol{a}+boldsymbol{d})+ )
( (boldsymbol{a}+mathbf{2 d})+ldots .+[boldsymbol{a}+(boldsymbol{n}-mathbf{1}) boldsymbol{d}]= )
( boldsymbol{n} / 2[2 boldsymbol{a}+(boldsymbol{n}-1) boldsymbol{d}] )
11
287. Use method of mathematical induction
2.75 +3.52.5 is divisible by 24 for all no
11
29The product of three consecutive natural numbers is divisible by This question has multiple correct options
( A cdot 3 )
B. 8
( c .6 )
D. 11
11
30For every natural number ( n, n(n+3) ) is
always :
A. multiple of 4
B. multiple of 5
c. even
D. odd
11
31Find ( m ) if the following equation holds true ( 1.2+2.3+3.4+. .+n(n+1)= )
( frac{1}{m} n(n+1)(n+2) )
11
32For every positive integer ( n, frac{n^{7}}{7}+frac{n^{5}}{5}+ ) ( frac{2 n^{3}}{3}-frac{n}{105} ) is
A. an integer
B. a rational number
c. a negative real number
D. an odd integer
11
33Prove using PMI: ( frac{1}{1.2 .3}+frac{1}{2.3 .4}+ )
( frac{1}{mathbf{3 . 4 . 5}}+ldots+frac{1}{n(n+1)(n+2)}= )
( frac{n(n+3)}{4(n+1)(n+2)} )
11
34The value of ( frac{1^{2}}{1.3}+frac{2^{2}}{3.5}+cdots+ )
( frac{n^{2}}{(2 n-1)(2 n+1)} ) is
A ( cdot frac{n(n+1)}{2(2 n+1)} )
B. ( frac{n(n-1)}{2(2 n-1)} )
c. ( frac{n^{2}(n-1)^{2}}{2(2 n+1)} )
D. none of these
11
35Let ( P(n)=n(n+1) ) is an even
number, then which of the following
satisfy ( boldsymbol{P}(boldsymbol{n}) )
A ( . P(3) )
B. ( P(100) )
c. ( P(50) )
D. All of these
11
36( sqrt{mathbf{2}}+sqrt{mathbf{2}}+sqrt{mathbf{2}}+ldots+boldsymbol{n} ) terms
( mathbf{2} cos frac{boldsymbol{pi}}{mathbf{2}^{n+1}}, boldsymbol{n} boldsymbol{epsilon} boldsymbol{N} )
11
37Prove that ( 3 n+1>3(n+1) )11
38Prove that ( 2.7^{n}+3.5^{n}-5 ) is divisible
by 24 for all ( n in N )
11
3928. Prove that
(2003
200-)
11
40Let ( p geq 3 ) be an integer and ( alpha, beta ) be the
roots of ( x^{2}-(p+1) x+1=0 . ) Using mathematical induction show that
( boldsymbol{alpha}^{n}+boldsymbol{beta}^{n} )
is an integer
11
41(1996 – 3 MUU
21. Let 0<A,<tt for i=1,2 …., n. Use mathematical induction to
prove that
(A + Azt….. + An)
sin A, + sin A, … + sin A s n sin
n
where 21 is a natural number.
{You may use the fact that
p sin x +(1-p) sin y < sin (px + (1-P)y],
where 0 < p < 1 and 0 <x, y < it.} (1997 – 5 Marks)
11
42Prove that ( 1^{2}+2^{2}+3^{2}+ldots .+n^{2}>frac{n^{2}}{3} )
for all ( n in N ) using principle of
mathematical induction
11
43If ( x_{1} x_{2} x_{3} dots x_{n}=1left(x_{1}>0, i=right. )
( 1,2, dots . n), ) prove that ( x_{1}+x_{2}+ )
( dots . . x_{n} geq n(n geq 2) )
11
44Prove by induction:
( x^{n}-y^{n} ) is divisible by ( x+y ) when ( n ) is
even
11
45Using mathematical induction prove
that:
[
frac{frac{1}{1.2 .3}+frac{1}{2.3 .4}+frac{1}{3.4 .5}+ldots . .+}{n(n+1)(n+2)}=frac{n(n+3)}{4(n+1)(n+2)}
]
11
46(1983- *
Use mathematical Induction to prove : If n is any odd
positive integer, then n(n2 – 1) is divisible by 24.
(1983 – 2 Marks)
11
47Let ( P(n): n^{2}+n ) is an odd integer
( boldsymbol{P}(boldsymbol{k}) Rightarrow boldsymbol{P}(boldsymbol{k}+1) ) is true
Then ( P(n) ) is true for all
( mathbf{A} cdot n>2 )
B ( . n>1 )
( c cdot n )
D. none of these
11
48Show that ( n(n+1)(2 n+1) ) is multiple
of 6 for every natural number ( n )
11
49Wut un
e an integer and a, ß be the roots of
Let p > 3 be an integer and a, ß be the
@+1)x+1=0 using mathematical induction show that
a” +B”.
(1) is an integer and (ii) is not divisible by p
(1002
A
11
50If ( a_{1}, a_{2}, a_{3}, dots, a_{n} ) are in A.P. and ( a_{i}>0 ) for all i, then show that ( frac{1}{a_{1} a_{2}}+frac{1}{a_{2} a_{3}}+ )
( ldots+frac{1}{a_{n-1} a_{n}}=frac{n-1}{a_{1} a_{n}} )
11
51Let ( boldsymbol{P}(boldsymbol{n}) ) be the statement ( boldsymbol{2}^{boldsymbol{n}}2 )
c. all ( n>3 )
D. all ( n<3 )
11
52Using the principle of mathematical induction prove that ( 2+4+6+ldots+ )
( mathbf{2} boldsymbol{n}=boldsymbol{n}^{2}+boldsymbol{n} )
11
53Prove the following by using the principle of mathematical induction for
all ( n in: )
( 1+frac{1}{(1+2)}+frac{1}{(1+2+3)}+dots dots dots+ )
( frac{1}{1+2+3+ldots . n)}=frac{2 n}{(n+1)} )
11
54Prove that 7 is a factor of ( 2^{3 n}-1 ) for all
natural numbers n.
11
55( forall boldsymbol{n} in boldsymbol{N}, mathbf{1}+frac{mathbf{1}}{sqrt{mathbf{2}}}+frac{mathbf{1}}{sqrt{mathbf{3}}}+ldots ldots+frac{mathbf{1}}{sqrt{boldsymbol{n}}} ) is
A ( cdot sqrt{n} )
( mathbf{B} cdot leq sqrt{n} )
( c cdot>sqrt{n} )
D. none of these
11
56Prove the following by using the principle of mathematical induction for
all ( boldsymbol{n} in boldsymbol{N}: mathbf{1 . 2}+mathbf{2 . 2}^{mathbf{2}}+mathbf{3 . 2}^{mathbf{2}}+ldots ldots . )
( boldsymbol{n} cdot boldsymbol{2}^{n}=(boldsymbol{n}-mathbf{1}) mathbf{2}^{n+mathbf{1}}+mathbf{2} )
11
57Prove the following by using the principle of mathematical induction for
all ( boldsymbol{n} in boldsymbol{N} )
( left(1+frac{3}{1}right)left(1+frac{5}{4}right)left(1+frac{7}{9}right) dots dots(1+ )
( (n+1)^{2} )
11
58By mathamatical induction ( nleft(n^{2}-1right) ) is divisible by
A . 19
B . 23
( c cdot 24 )
D . 29
11
59Find ( m ) if the following equation holds true ( 1.3+2.4+3.5+ldots+n(n+2)= )
( frac{1}{m} n(n+1)(2 n+7) )
11
60Prove that ( (1 times 2 times 5)+(2 times 3 times 7)+ )
( (3 times 4 times 9)+ldots . ) is
( frac{n(n+1)(n+2)(3 n+7)}{6} )
11
615.
If an=
7+ … … haing n radical signs then by
methods of mathematical induction which is true 
(a) an > 7 Vn21 (b) an <7 Vn21
(c) an < 4 vnzl (d) an <3 V nz1
If
ignogitia the first nocotine torm in the expansion of
11
62Prove the following by mathematical
induction ( 1+2+3+cdots+n< )
( frac{1}{8}(2 n+1)^{2} )
11
63If ( forall boldsymbol{m} in boldsymbol{N}, ) then ( mathbf{1 1}^{m+2}+mathbf{1 2}^{2 m-1} ) is
divisible by
A . 121
B. 132
c. 133
D. None of these
11
6429. A coin has probability p of showing head when tossed. It is
tossed n times. Let p denote the probability that no two (or
more) consecutive heads occur. Prove that p=1, p=1-p2
and p =(1-p). Pn-1+p(1 – p) Pn-2 for all n>3.
Prove by induction on n, that Pn = Aa” + BB” for all n>1,
where a and B are the roots of quadratic equation
x2_(1 – p)x-p (1–p)=0 and A=P+B-1 B-pº+a-1
aß-B2
aß-a26
11
65Prove that ( 2.7^{n}+3.5^{n}-5 ) is divisible
by ( 24, ) for all ( n in N )
11
66Prove by mathematical induction,
( mathbf{1}^{2}+mathbf{2}^{2}+mathbf{3}^{2}+ldots+boldsymbol{n}^{2}= )
( frac{boldsymbol{n}(boldsymbol{n}+mathbf{1})(boldsymbol{2} boldsymbol{n}+mathbf{1})}{boldsymbol{6}} )
11
67Observing that ( 1^{3}=1,2^{3}=3+5 )
( 3^{3}=7+9+11,4^{3}=13+15+17+19 )
Find a general formula for the cube of natural number n and prove it by the principle of mathematical induction
11
68Prove the following by using the first
principle ( mathbf{1}+mathbf{3}+mathbf{3}^{2}+ldots ldots+mathbf{3}^{n-1}=frac{left(mathbf{3}^{n}-mathbf{1}right)}{mathbf{2}} )
11
6912.
+
n7 n5
Prove that +
7 5
positive integer n.
2n3
3
n
105
he is an integer for every
(1990 – 2 Marks
11
70Prove that ( n^{3}+3 n^{2}+5 n+3 ) is
divisible by 3 for any natural ( n )
11
71( f A=left[begin{array}{lll}1 & 1 & 1 \ 1 & 1 & 1 \ 1 & 1 & 1end{array}right], ) prove that
( boldsymbol{A}^{n}=left[begin{array}{ccc}mathbf{3}^{n-1} & mathbf{3}^{n-1} & mathbf{3}^{n-1} \ mathbf{3}^{n-1} & mathbf{3}^{n-1} & mathbf{3}^{n-1} \ mathbf{3}^{n-mathbf{1}} & mathbf{3}^{n-mathbf{1}} & mathbf{3}^{n-1}end{array}right], boldsymbol{n} in N )
11
7216 divides ( n^{4}+4 n^{2}+11, ) if ( n ) is an odd
integer.
A. True
B. False
11
73For every positive integral value of ( n )
( mathbf{3}^{n}>n^{3} ) when
( A cdot n>2 )
в. ( n geq 3 )
c ( . n geq 4 )
( D cdot n<4 )
11
74( forall n in N, 3^{3 n}-26^{n} ) is divisible by
A .24
B. 64
c. 17
D. none of these
11
75Let ( P(n): a^{n}+b^{n} ) such that ( a, b ) are
even, then ( p(n) ) will be divisible by ( a+b )
if
( mathbf{A} cdot n>1 )
B. ( n ) is odd
( mathbf{c} cdot n ) is even
D. none of these
11
76If ( boldsymbol{x}^{n}-mathbf{1} ) is divisible by ( boldsymbol{x}-boldsymbol{k}, ) then the
least positive integral value of ( k ) is
A . 1
B . 2
( c .3 )
D. 4
11
77State whether following statement is
true or false.

By using the principle of mathematical induction we can proove that
( boldsymbol{n}(boldsymbol{n}+1)(boldsymbol{n}+mathbf{5}) ) is a multiple of ( boldsymbol{3} )
A. True
B. False

11
78For ( boldsymbol{n} in boldsymbol{N}, boldsymbol{x}^{n+1}+(boldsymbol{x}+mathbf{1})^{2 n-1} ) is
divisible by
( A )
B. ( x+1 )
c. ( x^{2}+x+1 )
D. ( x^{2}-x+1 )
11
79Let ( boldsymbol{x}>-mathbf{1}, ) then statement ( boldsymbol{p}(boldsymbol{n}):(mathbf{1}+ )
( boldsymbol{x}^{n}>1+boldsymbol{n} boldsymbol{x}, ) where ( boldsymbol{n} in boldsymbol{N} ) is true for
A. For all ( n epsilon N )
B. For all ( n>1 )
c. For all ( n>1 ), provided ( x neq 0 )
D. For all ( n>2 )
11
80prove the inequalities ( (n !)^{2} leq )
( boldsymbol{n}^{n}(boldsymbol{n} !)<(2 boldsymbol{n}) ! ) for all positive integers
( mathbf{n} )
11
81Let S(K) =1+3+5…+(2K-1) = 3+K2 . Then which of
the following is true

(a) Principle of mathematical induction can be used to
prove the formula
(b) S(K)= S(K +1)
(c) S(K) S(K +1)
(d) S(1) is correct
11
82For natural number ( n, 2^{n}(n-1) !<n^{n} )
if
( mathbf{A} cdot n2 )
c ( . n geq 2 )
D. Never
11
83Statement-I: For every natural number ( boldsymbol{n} geq 2, frac{1}{sqrt{1}}+frac{1}{sqrt{2}}+ldots ldots+frac{1}{sqrt{n}}>sqrt{boldsymbol{n}} )
Statement-2: For every natural number ( boldsymbol{n} geq mathbf{2}, sqrt{boldsymbol{n}(boldsymbol{n}+mathbf{1})}<boldsymbol{n}+mathbf{1} )
A. Statement-1 is true, Statement-2 is true; Statement- is a correct explanation for Statement-
B. Statement-1 is true, Statement-2 is false.
c. Statement- – is false, statement-2 is true
D. Statement-1 is true, Statement-2 is true; Statement- is not a correct explanation for Statement-
11
84A student was asked to prove a statement by induction. He proved
(i) ( P(5) ) is true and
(ii) Trutyh of ( P(n) Rightarrow ) truth of ( p(n+1), n in )
( mathbf{N} )
On the basis of this, he could conclude that ( P(n) ) is true for
A. no ( n in N )
B. all n in N
c. all ( n geq 5 )
D. None of these
11
85Prove that ( n^{2}-n ) is divisible by 2 for
every positive integer.
11
86Prove that ( 3^{n+1}>3(n+1) )11
87Let ( P(n): n^{2}+n ) is an odd integer. It
is seen that truth of ( P(n) Rightarrow ) the truth of ( P(n+1) . ) Therefore, ( P(n) ) is true for all
( A cdot n>1 )
B.
( c cdot n>2 )
D. None of these
11
88If ( a, b ) and ( n ) are natural numbers then
( a^{2 n-1}+b^{2 n-1} ) is divisible by
A ( cdot a+b )
в. ( a-b )
( mathbf{c} cdot a^{3}+b^{3} )
D. ( a^{2}+b^{2} )
11
89UutuoU
U
……”
Using mathematical induction prove that for every integer
n 2 1, (32n-1) is divisible by 2n+2 but not by 2nts.
(1996 – 3 Marks)
11
90If ( boldsymbol{P}(boldsymbol{n})=mathbf{1}+mathbf{2}+mathbf{3}+cdots+boldsymbol{n} ) is a
perfect square, ( N ) is less than 100 , then
possible values of ( n ) is/are
A. only 1
B. 1 and 8
c. only 8
D. 1,8,49
11
91Prove the assertions of the following
problems Prove that the expression ( n^{3}-n ) is
divisible by 24 for any odd ( n )
11
92The integer next above ( (sqrt{3}+1)^{2 n} )
contains
( mathbf{A} cdot 2^{n+1} ) as a factor
B. ( 2^{n+2} ) as a factor
( mathbf{c} cdot 2^{n+3} ) as a factor
D. ( 2^{n} ) as a factor
11
93If ( P(n) ) is the statement ( n^{2}-n+41 ) is
prime, prove that ( boldsymbol{P}(mathbf{1}), boldsymbol{P}(2) ) and ( boldsymbol{P}(boldsymbol{3}) )
are true.
Prove also that ( P(41) ) is not true.
11
94( mathbf{1} . mathbf{3}+mathbf{3 . 5}+mathbf{5 . 7}+ldots ldots ldots . .+(mathbf{2 n}- )
1) ( (2 n+1)=frac{nleft(4 n^{2}+6 n-1right)}{3} ) is true
for
A. Only natural number ( n geq 4 )
B. Only natural numbers 3 ( leq n leq 10 )
c. All natural numbers n
D. None
11
95The number ( 4^{n}+15 n-1 ) is a
multiple of 9 for any natural ( n )
A. True
B. False
11
96Prove by induction:
( 2+2^{2}+2^{3}+ldots ldots .+2^{n}=2left(2^{n}-1right) )
11
97Using the Mathematical induction,
show that for any number ( n geq 2 )
[
begin{array}{c}
frac{1}{1+2}+frac{1}{1+2+3}+frac{1}{1+2+3+4}+ \
dots+frac{1}{1+2+3+ldots+n}=frac{n-1}{n+1}
end{array}
]
11
98Prove that ( 1+3+5+ldots .+(2 n- )
1) ( =n^{2} )
11
99Let ( a, b, c ) and ( d ) be any four real
numbers. Then, ( a^{n}+b^{n}=c^{n}+d^{n} ) holds
for any natural number ( n ), if
(This question has some ambiguity, but appeared in WBJEE 2015 exam)
A ( . a+b=c+d )
В . ( a-b=c-d )
c. ( a+b=c+d, a^{2}+b^{2}=c^{2}+d^{2} )
D . ( a-b=c-d, a^{2}-b^{2}=c^{2}-d^{2} )
11
100The inequality, ( P(n)=n !>2^{n} ) is true
for
( mathbf{A} cdot n geq 4 )
в. ( n>1 )
c ( . n>2 )
D. ( forall n in N )
11
101Using the principle mathematical induction, prove that ( forall n epsilon N ) ( boldsymbol{I}_{n}=int_{0}^{pi / 2} cos ^{n} boldsymbol{x} sin boldsymbol{n} boldsymbol{x} boldsymbol{d} boldsymbol{x} )
( =frac{1}{2^{n+1}}left[2+frac{2^{2}}{2}+frac{2^{3}}{3} ldots+frac{2^{n}}{n}right] )
11
102If ( n ) is an even number, then the digit in
the units place of ( 2^{2 n}+1 ) will be
( mathbf{A} cdot mathbf{5} )
B. 7
( c cdot 6 )
D.
11
103If ( boldsymbol{y}=frac{log boldsymbol{x}}{boldsymbol{x}}, ) then prove by
mathematical induction
( boldsymbol{y}_{n}= )
( frac{(-1)^{n}(n !)}{x^{n+1}}left[log x-1-frac{1}{2}-dots-frac{1}{n}right] )
11
104Show by the mathematical induction
that ( frac{1}{sin 2 x}+frac{1}{sin 4 x}+frac{1}{sin 2^{n} x}= )
( cot x-cot 2^{n} x )
11
105( forall n in N ; x^{2 n-1}+y^{2 n-1} ) is divisible by?
A. ( x-y )
B. ( x+y )
c. ( x y )
D. ( x^{2}+y^{2} )
11
106Prove the following by using the principle of mathematical induction for
all ( boldsymbol{n} in boldsymbol{N}: mathbf{1} . mathbf{2}+mathbf{2} . mathbf{3}+mathbf{3} . mathbf{4}+ldots . .+ )
( boldsymbol{n}(boldsymbol{n}+1)=left[frac{boldsymbol{n}(boldsymbol{n}+1)(boldsymbol{n}+boldsymbol{2})}{boldsymbol{3}}right] )
11
107Prove the following by using the principle of mathematical induction for all ( boldsymbol{n} in boldsymbol{N}: mathbf{1} . mathbf{2 . 3}+mathbf{2 . 3 . 4}+ldots ldots+ )
( boldsymbol{n}(boldsymbol{n}+mathbf{1})(boldsymbol{n}+mathbf{2})= )
( frac{boldsymbol{n}(boldsymbol{n}+mathbf{1})(boldsymbol{n}+mathbf{2})(boldsymbol{n}+mathbf{3})}{mathbf{4}} )
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10810. Using mathematical induction, prove that(1989- 3 Marks)
m Co “Cx + G “Cx-1 +……. MCK “Co = (m+n)Ck2
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109( mathbf{1}^{2}+mathbf{2}^{2}+ldots+boldsymbol{n}^{2}>frac{n^{3}}{3}, boldsymbol{n} in mathbf{N} )11
11018.
Ifx is not an integral multiple of 2īt use mathematical inducting
to prove that :
(1994 – 4 Marko
n+1
COS X + cos2x +…….. + cos nx = cos
nx
-X sin –
x
2
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111For all positive integers ( n, P(n) ) is true
and ( 2^{n-2}>3 n ), then which of the
following is true?
( mathbf{A} cdot P(3) ) is true
( mathbf{B} cdot P(5) ) is true
C. If ( P(m) ) is true then ( P(m+1) ) is also true.
D. If ( P(m) ) is true then ( P(m+1) ) is not true.
11
112Given ( (1+x)^{n} geq(1+n x), ) for all
natural number ( n, ) where ( x>-n . ) What
will be the value of ( n )
11
113Use mathematical induction to prove:
( frac{1}{1 cdot 3}+frac{1}{3 cdot 5}+frac{1}{5 cdot 7}+dots+ )
( frac{1}{(2 n-1)(2 n+1)}=frac{n}{2 n+1} )
11
114For any integer ( n geq 1 ), the sum ( sum_{k=1}^{n} k(k+2) ) is equal to
A ( cdot frac{n(n+1)(n+2)}{6} )
B. ( frac{n(n+1)(2 n+1)}{6} )
c. ( frac{n(n+1)(2 n+7)}{6} )
D. ( frac{n(n+1)(2 n+9)}{6} )
11
115If ( 33 ! ) is divisible by ( 2^{n}, ) then find the
maximum value of ( n )
11
116Prove the following using the principle
of mathematical induction for all ( boldsymbol{n} in boldsymbol{N} )
( 1+3+3^{2}+ldots ldots . .+3^{n-1}=frac{left(3^{n}-1right)}{2} )
11
117The smallest positive integer for which
the statement ( 3^{n+1}<4^{n} ) holds is
( mathbf{A} cdot mathbf{1} )
B . 2
( c .3 )
D. 4
11
118If ( p ) is a prime number, then ( n^{p}-n ) is
divisible by ( p ) for all ( n, ) where
A ( . n in N )
B. ( n ) is odd natural number
c. ( n ) is even natural number
D. ( n ) is not a composite number
11
119Let ( boldsymbol{P}(boldsymbol{n}) ) be the statement ( ” boldsymbol{3}^{boldsymbol{n}}>boldsymbol{n}^{prime prime} . ) If
( P(n) ) is true, ( P(n+1) ) is true.
A. True
B. False
11
120Assertion
( (A): ) If ( A=(300)^{600}, B=600 !, C= )
( (200)^{600}, ) then ( A>B>C )
Reason
( (mathrm{R}):left(frac{n}{2}right)^{n}>n !>left(frac{n}{3}right)^{n} ) for ( n>6 )
A. Both (A) & (R) are individually true & (R) is correct explanation of (A).
B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A)
c. (A) is true but (R) is false.
D. (A) is false but (R) is true.
11
121Let ( f(n) ) equals to the sum of the cubes
of three consecutive natural numbers.
( f(n) ) leaves the remainder zero when
divided by
A . 11
B. 9
c. 99
D. none of these
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122( operatorname{Let} boldsymbol{P}(boldsymbol{n})=mathbf{5}^{n}-mathbf{2}^{n} . boldsymbol{P}(boldsymbol{n}) ) is divisible by
( 3 lambda ) where ( lambda ) and ( n ) both are odd positive integers, then the least value of ( n ) and ( lambda )
will be
A . 13
B. 11
( c cdot 1 )
D. 5
11
1237 is a factor of ( 2^{3 n}-1 ) for all natural
numbers n.
A. True
B. False
11
12426. Let a, b, c be positive real numbers such that b2 – 4ac >0
and let a, = c. Prove by induction that
2.
aa
an+1
tis well – defined and
162 – 2ala, + a2 +…+an)
3.
Q,-,<en for all n=1, 2, … (Here, 'well – defined' means
that the denominator in the expression for an+, is not zero.)
(2001 – 5 Marks)
11
125OLULILU
Prove by mathematical induction that – (1987 – 3 M
(2n)! 1
for all positive Integers n.
22n (n!) 2 (3n+1) 1/2
11
126Prove the following by using the principle of mathematical induction for
all ( boldsymbol{n} in boldsymbol{N}: frac{mathbf{1}}{mathbf{2 . 5}}+frac{mathbf{1}}{mathbf{5 . 8}}+frac{mathbf{1}}{mathbf{8 . 1 1}}+dots dots+ )
( frac{1}{(3 n-1)(3 n+2)}=frac{n}{(6 n+4)} )
11
127(2000 – 6 Marks)
25. For every positive integer n, prove that
(4n+1) < Vn+ Vn+1 < 4n+2. Hence or otherwise,
prove that [Vn+ ✓(n+1)] = [V4n+1], where [x] denotes
the greatest integer not exceeding x. (2000 – 6 Marks
11
128If ( boldsymbol{A}=left[begin{array}{cc}cos theta & sin theta \ -sin theta & cos thetaend{array}right] ) then show
that for all the positive integers ( n A^{n}= ) ( left[begin{array}{cc}cos n theta & sin n theta \ -sin n theta & cos n thetaend{array}right] )
11
129Prove using PM ( left(1+frac{3}{1}right)left(1+frac{5}{4}right)left(1+frac{7}{9}right) dotsleft(1+frac{(2 n+1)}{n^{2}}right)= )
( (n+1)^{2} )
11
13016. Using mathematical induction, prove that
tan?(1/3) + tan ?(1/7) +…. tan-1{1/(n2 +n+1)}
= tan-‘{n/(1+2)}
(1993 – 51
11
131Let ( P(n) ) be a statement and ( P(n)= ) ( boldsymbol{P}(boldsymbol{n}+mathbf{1}) forall boldsymbol{n} in boldsymbol{N}, ) then ( boldsymbol{P}(boldsymbol{n}) ) is true for
what values of ( n ? )
A. For all ( n )
B. For all ( n>1 )
c. For all ( n>m, m ) being a fixed positive integer
D. Nothing can be said
11

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