Answer
$5\left(1-\left(\dfrac{3}{5}\right)^n\right)$
Work Step by Step
We are given the sum:
$2+\dfrac{6}{5}+\dfrac{18}{25}+...+2\left(\dfrac{3}{5}\right)^{n-1}$
Consider the geometric sequence:
$a_1=2$
$r=\dfrac{3}{5}$
The given sum represents the sum $S_n$ of the first $n$ terms. Determine this sum:
$S_n=a_1\dfrac{1-r^n}{1-r}=2\cdot\dfrac{1-\left(\dfrac{3}{5}\right)^n}{1-\dfrac{3}{5}}$
$=2\cdot \dfrac{1-\left(\dfrac{3}{5}\right)^n}{\dfrac{2}{5}}$
$=5\left(1-\left(\dfrac{3}{5}\right)^n\right)$
