Question

Show that \(9^{ n+1} -8n -9\) is divisible by \(64\), whenever n is a positive integer.

Answer 1

To prove:

9ⁿ⁺¹ − 8n − 9 is divisible by 64, for all positive integers n.

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Step 1: Rewrite the expression

9ⁿ⁺¹ − 8n − 9 = 9·9ⁿ − 9 − 8n

= 9(9ⁿ − 1) − 8n

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Step 2: Use binomial expansion

9 = 1 + 8

So,

9ⁿ = (1 + 8)ⁿ

= 1 + n·8 + ⁿC₂8² + …

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Step 3: Subtract 1

9ⁿ − 1 = 8n + 64k, where k is an integer

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Step 4: Substitute back

9(9ⁿ − 1) − 8n = 9(8n + 64k) − 8n

= 72n + 576k − 8n

= 64n + 576k

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Step 5: Factor 64

64n + 576k = 64(n + 9k)

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Conclusion:

Since n + 9k is an integer,

9ⁿ⁺¹ − 8n − 9 is divisible by 64.