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**Prove It** Well then do as you were instructed by by Manhole, his psychic powers must be a little more powerful and finely tuned than mine...

$\displaystyle \displaystyle \begin{align*} \sum_{n = 0}^{\infty}\frac{3^n + 5}{4^n} &= \sum_{n = 0}^{\infty}\left(\frac{3^n}{4^n} + \frac{5}{4^n}\right) \\ &= \sum_{n = 0}^{\infty}\left(\frac{3}{4}\right)^n + 5\sum_{n = 0}^{\infty}\left(\frac{1}{4}\right)^n \end{align*}$

The first is an infinite geometric series with $\displaystyle \displaystyle \begin{align*} a = 1, r = \frac{3}{4} \end{align*}$ and the second is an infinite geometric series with $\displaystyle \displaystyle \begin{align*} a = 1, r = \frac{1}{4} \end{align*}$.