Show that 9ⁿ⁺¹ – 8n – 9 is divisible by 64, whenever n is a positive integer.

In order to show that 9ⁿ⁺¹ – 8n – 9 is divisible by 64,

we have to prove that

9 ⁿ⁺¹ – 8n – 9 = 64 k, where k is some natural number

Now,

9ⁿ⁺¹ = (1+8)ⁿ⁺¹

We know that-

putting a =1, b = 8, and n = n+1

Hence,

Taking out (8)² from right side, we get-

where is a natural number

Thus, 9ⁿ⁺¹ – 8n – 9 is divisible by 64.

Hence Proved.