Answer
Geometric
Sum: $\approx 1.99999$
Work Step by Step
We are given the sequence:
$\left\{\left(\dfrac{2}{3}\right)^n\right\}$
Compute the ratio between two consecutive terms:
$\dfrac{a_{k+1}}{a_k}=\dfrac{\left(\dfrac{2}{3}\right)^{k+1}}{\left(\dfrac{2}{3}\right)^k}=\dfrac{2}{3}$
As the ratio between any consecutive terms is constant, the sequence is geometric.
Its elements are:
$a_1=\left(\dfrac{2}{3}\right)^1=\dfrac{2}{3}$
$r=\dfrac{2}{3}$
We determine the sum of the first 50 terms:
$S_n=a_1\cdot\dfrac{1-r^n}{1-r}$
$S_{50}=\dfrac{2}{3}\cdot\dfrac{1-\left(\dfrac{2}{3}\right)^{50}}{1-\dfrac{2}{3}}\approx 1.99999$
