Let the given statement be P(n), as

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

First, we check if it is true for n = 1,

P(1):3⁴ - 8 - 9 = 81 - 17 = 64 = 8(8);

∴ It is true for n = 1.

Now we assume that it is true for some positive integer k, such that

P(k):3^{2k + 2} - 8k - 9 = 8m where m ∈ N.

3^{2k + 2} = 8k + 9 + 8m ………….(1)

We shall prove that P(k + 1)is true,

P(k + 1):3^{2k + 4} - 8(k + 1) - 9

⇒ 3^{2k + 2}.3² - 8k - 8 - 9

⇒ (8k + 9 + 8m)9 - 8k - 17 From equation(1)

⇒ 64k + 72m + 81 - 17

⇒ 64k + 72m + 64

⇒ 8(8k + 9m + 8)

We proved that P(k + 1) is true.

Hence by principle of mathematical induction it is true for all n ∈ N.