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

Let the given statement be P(n), i.e.,

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

It can be observed that P(n) is true for n = 1 since 3^{2 × 1 + 2} – 8 × 1 – 9 = 64, which is divisible by 8.

Let P(k) be true for some positive integer k, i.e.,

3^{2k + 2} – 8k – 9 is divisible by 8.

∴3^{2k + 2} – 8k – 9 = 8m; where m ∈ N … (1)

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

Consider

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.