Question

Question 22 Prove the following by using the principle of mathematical induction for all n∈N: 3^(2n + 2) – 8n – 9 is divisible by 8.

Class X1 - Maths -Principle of Mathematical Induction Page 95

Answer 1

3^(2n+2)-8n -9 is divisible by 8.
Let P(n):3^(2n+2)-8n-9 is divisible by 8.

step1:- for n = 1
p(1):3^(2×1+2)-8×1 -9
P(1) = 81 - 8 - 9
= 64
it is clear that p(1) is divisible by 8.

Step2:- for n = k
P(k): 3^(2k+2)-8k-9 is divisible by 8.
Let, 8L = 3^(2k+2)-8k-9 ----(1)

step3:- for n=k+1
P(k+1)= 3^{2(k+1)+2} -8(k+1)-9
= 3^(2k+2+2) -8k -8 -9
= 3^(2k+2).3² - 8k-17
= 3²(8L+8k+9) -8k-17
=72L + 72k + 81 - 8k-17
= 72L + 64k + 64
= 8(9L + 8k + 8)
it's clear that it is divisible by 8.
here, p(k+1) is true when p(k+1) is true. Hence, form the principle of mathematical induction, statement is true.