using binomial therorem, 3^{2n+2}-8n-9 is divisible by 64, n belongs to N

In order to show that 3^{2}^{n}^{+2} - 8*n* - 9 is divisible by 64, it has to be proved that,

, where *k* is some natural number and

3^{2}^{n}^{+2} = 3^{2.(}^{n}^{+1)} = 9^{n}^{+1} ..... (1)

By Binomial Theorem,

For *a* = 8 and *m* = *n* + 1, we obtain

⇒ 3^{2}^{n}^{+2} - 8*n* - 9 = 64k [using (1)]

Thus, 3^{2}^{n}^{+2} - 8*n* - 9 is divisible by 64, whenever *n* is a positive integer.

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