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| Math Central | Quandaries & Queries |
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Question from Jerry, a student: How would you prove that for any positive integer n, the value of the expression 3^(2n+2) - 8n -9 is divisible by 64. |
Jerry, we have two responses for you:
Hi Jerry.
The only thing tricky about this problem is that you are given the form for (n+1). One easy way to begin your solution would be to write the term for n, noting that 3(2n) = 9n. Note that the given expression for n+1 is just 9 times the expression for n plus something. It is your job to determine that something.
Chris Fisher.
Jerry,
At the induction step write 3(2n+4) - 8(n+1) -9 as
9×3(2n+2) - 9×8n + 8×8n - 9×9 + 64
and if you collect terms carefully you'll see the next
step.
Penny
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