<s>[INST] Infinite summation with multiple terms in denominator and variables in exponents I need to find $\sum\limits_{n=1}^{\infty} \frac{6^n}{2^{1 + 2 n} + 3^{1 + 2 n} - 5\cdot 6^n}$, but I don't know how to do infinite summation with multiple terms in the denomiator and variables in exponents. Can anybody give me a hint? Thanks :)
 [/INST] Hint
Let $x=2^n$ and $y=3^n$ to make
$$\frac{6^n}{2^{1 + 2 n} + 3^{1 + 2 n} - 5\times 6^n}=\frac{x y}{2 x^2-5 x y+3 y^2}=\frac{x y}{(2 x-3 y) (x-y)}$$ Now, as @person commented, using partial fractions
$$\frac{x y}{(2 x-3 y) (x-y)}=\frac{x}{y-x}-\frac{2 x}{3 y-2 x}=\frac{2^n}{3^n-2^n}-\frac{2^{n+1}}{3^{n+1}-2^{n+1}}$$ which seems to telescope.
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