Answer
$5\displaystyle \left[ 1-\left(\frac{3}{5}\right)^{n} \right]$
Work Step by Step
We can write the sum as $S=\displaystyle \sum_{k=1}^{n}2\cdot(\frac{3}{5})^{n-1}$
Apply
THEOREM: Sum of the First $n$ Terms of a Geometric Sequence
$S_{n}=\displaystyle \sum_{k=1}^{n}a_{1}r^{k-1}=a_{1}\cdot\frac{1-r^{n}}{1-r},\quad r\neq 0,1$
$=2\displaystyle \left[\frac{1-\left(\frac{3}{5}\right)^{n}}{1-\frac{3}{5}}\right]$
$=2\displaystyle \left[\frac{1-\left(\frac{3}{5}\right)^{n}}{\frac{2}{5}}\right]$
$=5\displaystyle \left[ 1-\left(\frac{3}{5}\right)^{n} \right]$
