# Using the principle of mathematical induction, prove each of the following for all n ϵ N:  (3^(2n+2) – 8n – 9) is divisible by 8.

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

(32n+2 – 8n – 9) is divisible by 8.

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To Prove:

(32n+2 – 8n – 9) is divisible by 8.

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

Let P(n): (32n+2 – 8n – 9) is divisible by 8

For n = 1 P(n) is true since

32n+2 – 8n – 9 = 32+2 - 8 x 1 - 9 = 81 - 17 = 64

Which is divisible of 8

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

= (32k+2 – 8k – 9) is divisible by 8

∴ 32k+2 – 8k – 9 = m x 8, 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] = 8×r, where r = 9m + 8k + 8 is a natural number

Therefore 32k+2 - 8k - 9 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.