Answer
$\displaystyle \frac{2^{n}-1}{4}$
Work Step by Step
$a_{1}=\displaystyle \frac{1}{4}$
$a_{2}=a_{1}\cdot 2$
$a_{3}=a_{1}\cdot 2^{2}$
$...$
The terms of the sum form a geometric sequence, $a_{1}=\displaystyle \frac{1}{4},\ r=2.$
THEOREM: Sum of the First $n$ Terms of a Geometric Sequence
$S_{n}=a_{1}\displaystyle \cdot\frac{1-r^{n}}{1-r},\quad r\neq 0,1$
$=\displaystyle \frac{1}{4}\left(\frac{1-2^{n}}{1-2}\right)$
$=-\displaystyle \frac{1}{4}(1-2^{n})$
$=\displaystyle \frac{1}{4}(2^{n}-1)$