Using binomial theoram ,show that 9ⁿ⁺¹-8n-9 is divisible by 64 ,whr n is a positive integer.

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

P(n): 9ⁿ⁺¹ - 8n - 9 = 3²ⁿ⁺² – 8n – 9 is divisible by 64.

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 64.

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

P(k): 3^{2k + 2} – 8k – 9 is divisible by 64.

∴3^{2k + 2} – 8k – 9 = 64m; where m ∈ N … (1)

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

Consider

Therefore, 3^{2k + 2} – 8k – 9 is divisible by 64.

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