SOLUTION: Prove that for any positive integer n, the value of the expression 3^(2n+2)-8n-9 is divisible by 64

SOLUTION: Prove that for any positive integer n, the value of the expression 3^(2n+2)-8n-9 is divisible by 64

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Question 889703: Prove that for any positive integer n, the value of the expression 3^(2n+2)-8n-9 is divisible by 64
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
We induct on n. n = 1 is trivial (3^4 - 8 - 9 = 64).

Assume

for some integer

. Then

.

We want to show that

. This is equivalent to showing that

. However, since we know that

, the statement we wish to prove is equivalent to

, which is true since the LHS equals

. The induction is complete and the statement holds for all positive integers n.
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