Solving Problem: Prove 3^{2n+2} - 8n -9 Divisible by 64

[recon]

Can someone point me in the right direction of solving the following problem:

Prove that for any postive integer n, the value of the expression 3^{2n+2} - 8n -9 is divisible by 64.

 

[honestrosewater]

I find when you're asked to prove that every member of some subset of the natural numbers has some property, induction usually works.

 

[matt grime]

9^{n+1}-8n-9

or equivalently

9^n-8n-1

what is the binomial expansion of 9^n when considering 9=8+1?

 

[honestrosewater]

Maybe I'm having a blonde moment, but doesn't a^{n + 1} - a = a(a^{n} - 1), making 9^{n + 1} - 9 - 8n = 9(9^{n} - 1) - 8n?

 

[matt grime]

ah, perhaps i ought to have been clearer: i wasn't say the epxressions are equal, but that if you prove one is divisible by 64 for all n, the other will be divislbe by 64 for all n (give or take a case when n=0). I let m=n+1 in the first, then relabelled n=m.

 

[recon]

I get it now. Thank you.