<s>[INST] A closed form for the sum $S = \frac {2}{3+1} + \frac {2^2}{3^2+1} + \cdots + \frac {2^{n+1}}{3^{2^n}+1}$ is ... A closed form for the sum $S = \frac {2}{3+1} + \frac {2^2}{3^2+1} + \cdots + \frac {2^{n+1}}{3^{2^n}+1}$ is $1 - \frac{a^{n+b}}{3^{2^{n+c}}-1}$, where $a$, $b$, and $c$ are integers. Find $a+b+c$.
 [/INST] Note that
$$\frac{2^{k+1}}{3^{2^{k+1}}-1} - \frac{2^{k}}{3^{2^{k}}-1} = \frac{2^{k}}{3^{2^{k}}-1} \left ( \frac{2}{3^{2^{k}}+1} - 1\right ) = -\frac{2^k}{3^{2^{k}}+1}$$
Thus the finite sum is a telescoping sum and depends only on the end terms of the series.
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