Answer
$S_{15}\approx 799.9756$
Work Step by Step
$\dfrac{a_{n+1}}{a_n}=\dfrac{800\left(\frac{1}{2}\right)^{n+1}}{800\left(\frac{1}{2}\right)^n}=\dfrac{1}{2}$
Since the ratio between consecutive terms is constant, the given sequence is geometric.
The first term is $a_{1}=800(\frac{1}{2})^{1}=400$.
The common ratio is $r=\frac{1}{2}$.
The formula for the sum of the first $n$ terms is:
$S_{n}=\frac{a_{1}(1-r^{n})}{1-r}$
$S_{15}=\frac{400\left[1-\left(\frac{1}{2}\right)^{15}\right]}{1-\frac{1}{2}}\approx799.9756$
