Answer
$S=8191.75$
Work Step by Step
Write the sum in the form: $\displaystyle \sum_{k=1}^{n}a_{1}r^{k-1}$
$S= \displaystyle \frac{1}{4}+\frac{2}{4}+\frac{2^{2}}{4}+\frac{2^{3}}{4}+\cdots+\frac{2^{14}}{4}=$
$=\displaystyle \frac{1}{4}(2^{0})+\frac{1}{4}(2^{1})+\frac{1}{4}(2^{2})+\frac{1}{4}(2^{3})+\cdots+\frac{1}{4}(2^{14})$
$=\displaystyle \sum_{n=1}^{15}\frac{1}{4}(2^{k-1})$
Using the calculator at desmos.com,
$S=8191.75$
