Infinite sum proof by induction

[Doom of Doom]

So I am trying to solve a problem.

Evaluate \sum_{k=1}^{\infty}\frac{6^{k}}{(3^{k}-2^{k})(3^{k+1}-2^{k+1})}.

Essentially, I've boiled it down to this, but I can't quite prove it:
\sum_{k=1}^{n}\frac{6^{k}}{(3^{k}-2^{k})(3^{k+1}-2^{k+1})}=\frac{6(3^{n}-2^{n})}{(3^{n+1}-2^{n+1})}
and the limit of this as n approaches infinity is 2.

I need to be able to prove that 3^{n+1}-2^{n+1}+6^{n-1}=(3^{n}-2^{n})^{2} in order for my induction hypothesis to work, and I'm having trouble for some reason. Help?

 

[d_leet]

I need to be able to prove that 3^{n+1}-2^{n+1}+6^{n-1}=(3^{n}-2^{n})^{2} in order for my induction hypothesis to work, and I'm having trouble for some reason. Help?

Perhaps beacuse it isn't true, consider n=3 then

3^(3+1)-2^(3+1)+6^(3-1)=81-16+36=101
whereas
(3^3-2^3)^2=(27-8)^2=(19)^2=361

 

[Doom of Doom]

Wow, I'm sorry. Perhaps that's why I was unable to do it. My notes are all over the place and I guess I kind of lost track of a lot of numbers somewhere.

I meant to have (3^{n-1}-2^{n-1})(3^{n+1}-2^{n+1})+6^{n-1}=(3^{n}-2^{n})^{2}

Now I'm able to prove it! Thanks anyway.