Mar Dionysius Senior Sec. School
Rishi Valley,
Mallappally, Kerala, India-689534
Paper : x1 mat Mathematical Induction
Q.1
Using the principle of mathematical induction, prove that n(n + 1) is a multiple of 2, for all n
Q.2
Solve by induction that
1 . 3 + 2 . 32 + 3 . 33 + ...+ n . 3n =
Q.3
(Marks : 3
(Marks : 3 )
Prove by induction that An= cosn , when it is given that A1= cos
> 2, the relations Am = 2Am - 1cos
N.
- Am - 2 hold true.
, A2 = cos2
and for every natural number m
(Marks : 3 )
Q.4
For a natural number m, let A be a set having m elements. Prove that p(A), the power set of A, has 2m
elements.
(Marks : 4 )
Q.5
If n is a positive integer, prove that
(Marks : 4 )
Q.6
Q.7
Prove that (3 +
)n + (3 -
)n is an integer, for all n
N.
(Marks : 4 )
For any natural number n > 1, prove
(Marks : 4 )
Q.8
Use induction to prove that n(n2 - 1) is divisible by 24, if n is any odd positive integer.
Q.9
Prove the identity: If x is not a multiple of 2
(Marks : 4 )
use mathematical induction to prove that cosx + cos2x + ... +
cosnx
Q.10
Show that 10n + 3 . 4n + 2+ 5 is divisible by 9 for each natural number n.
Q.11
Let u1 = 1, u2 = 1, un + 2 = un + 1 + un for n
1, use mathematical induction to shows that
for all n
Q.12
(Marks : 6 )
1.
(Marks : 6 )
Prove by mathematical induction that
(Marks : 6 )
Q.13
Prove by the method of induction that 7 + 77 + 777...
.
(Marks : 6 )
Q.14
Prove the identity: Let 0 < Ai <
sinAn
n
for i = 1, 2, ..., n, then sinA1 + sinA2 + ...
where n
1 is a natural number.
(Marks : 6 )
Q.15
For all positive integers n, prove that
Q.16
is an integer.
(Marks : 6 )
Using induction or otherwise, prove that for any non-negative integers m, n, r and k,
(Marks : 6 )