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