<s>[INST] Finding the sum of series $\sum_{n=0}^∞ \frac{2^n + 3^n}{6^n}$ I am being asked to find the sum of the following convergent series :
$$\sum_{n=0}^∞ \frac{2^n}{6^n} + \frac{3^n}{6^n}$$
Attempting to generalize from partial sums yields nothing of interest:
$s_1 = \frac{5}{6}$
$s_2 = \frac{5}{6} + \frac{13}{36} = \frac{43}{36}$
$s_3 = \frac{43}{36} + \frac{35}{216} = \frac{293}{216}$
$s_4 = \frac{293}{216} + \frac{97}{1296} = \frac{1855}{1296} $
I do not see a pattern here...
How must I proceed to find the sum of this series?
 [/INST] It is just the sum of two geometric series in disguise.
$$
\begin{aligned}
\underset{i=0}{\overset{\infty}{\sum}}\frac{2^{n}+3^{n}}{6^{n}}&=\underset{i=0}{\overset{\infty}{\sum}}\frac{2^{n}}{6^{n}}+\frac{3^{n}}{6^{n}}\\
&=\underset{i=0}{\overset{\infty}{\sum}}(1/3)^{n}+\underset{i=0}{\overset{\infty}{\sum}}(1/2)^{n}\\
&=\frac{1}{1-\frac{1}{3}}+\frac{1}{1-\frac{1}{2}}\\
&=\frac{3}{2}+2\\
&=\frac{7}{2}.
\end{aligned}
$$
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