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.