Using the principle of mathematical induction, prove each of the following for all n ϵ N:

(3²ⁿ⁺² – 8n – 9) is divisible by 8.

To Prove:

Let us prove this question by principle of mathematical induction (PMI) for all natural numbers

Let P(n):

For n = 1 P(n) is true since

which is divisible of 8

Assume P(k) is true for some positive integer k , ie,

=

, where m ∈ N …(1)

We will now prove that P(k + 1) is true whenever P( k ) is true

Consider ,

[ Adding and subtracting 8k + 9 ]

= 9(8m) + 72k + 81 -8k-17 [ Using 1 ]

= 9(8m) + 64k + 64

= 8(9m + 8k + 8)

= 8×r , where r = 9m + 8k + 8 is a natural number

Therefore is a divisible of 8

Therefore, P (k + 1) is true whenever P(k) is true

By the principle of mathematical induction, P(n) is true for all natural numbers, ie, N.

Hence proved.