Answer
$$-\frac{4}{15} $$
Work Step by Step
Given $$\sum_{n=0}^{n=\infty} \frac{3(-2)^{n}-5^{n}}{8^{n}}=\sum_{n=0}^{n=\infty} 3 \cdot\left(\frac{-2}{8}\right)^{n}-\sum_{n=0}^{n=\infty}\left(\frac{5}{8}\right)^{n}$$
Since the series is a geometric series with $|r_1|= \frac{2}{8}<1$ and $|r_2|= \frac{5}{8}<1$, then the series converges and has the sum
\begin{align*}
S&=S_1-S_2\\
&=\frac{a_1}{1-r_1}-\frac{a_1}{1-r_2}\\
&=\frac{3}{1-\frac{-2}{8}}-\frac{1}{1-\frac{5}{8}}\\
&=-\frac{4}{15}
\end{align*}
