Proof by induction stuck on a sum help needed

**ssadi** · Dec 16th 2008, 06:58 AM

Code:

Given that m is an odd positive integer, prove that 




 is divisible by 32 for all such values of m.

I made it divisible by 8:
$((k+2)^2+3)((k+2)^2+15)-((k^2+3)(k^2+15))<br /> =8 (k^3+3k^2+13k+11)$
But I could not make it divisible by 32. A little help please, I feel I was very close to the solution.
My method is right, right?

**Moo** · Dec 16th 2008, 07:06 AM

Hello,

Originally Posted by ssadi

Code:

Given that m is an odd positive integer, prove that 




 is divisible by 32 for all such values of m.

I made it divisible by 8:
$((k+2)^2+3)((k+2)^2+15)-((k^2+3)(k^2+15))<br /> =8 (k^3+3k^2+13k+11)$
But I could not make it divisible by 32. A little help please, I feel I was very close to the solution.
My method is right, right?

You can use the fact that k is odd, that is to say $k=2n+1$

It'll give you what you want, but the method you use looks very long to me...
Dunno if there's another way (thinking on it)

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