Using binomial theorem prove that (3²ⁿ⁺²- 8n - 9) is divisible by 64,where n is a positive integer. Please give me the solution of this as soon as possible. Thank u.

In order to show that 3²ⁿ⁺²- 8n- 9 is divisible by 64, it has to be proved that,

, wherekis some natural number and

3²ⁿ⁺²= 3^2.(n+1)= 9ⁿ⁺¹..... (1)

By Binomial Theorem,

Fora= 8 andm=n+ 1, we obtain

⇒3²ⁿ⁺²- 8n- 9 = 64k[using (1)]

Thus,3²ⁿ⁺²- 8n- 9 is divisible by 64, whenevernis a positive integer.

In order to show that 3²ⁿ⁺²- 8n- 9 is divisible by 64, it has to be proved that,

, wherekis some natural number and 3²ⁿ⁺²= 3^2.(n+1)= 9ⁿ⁺¹..... (1)

By Binomial Theorem,

Fora= 8 andm=n+ 1, we obtain

⇒3²ⁿ⁺²- 8n- 9 = 64using (1)]

Thus,3²ⁿ⁺²- 8n- 9 is divisible by 64, whenevernis a positive integer.

thank u vidyullata :)