Answer
$S_{10}\approx -666.597$
Work Step by Step
$\dfrac{a_{n+1}}{a_n}=\dfrac{-1000\left(\frac{2}{5}\right)^{n+1}}{-1000\left(\frac{2}{5}\right)^n}=\dfrac{2}{5}$
Since the ratio between consecutive terms is constant, the given sequence is geometric.
The first term is $a_{1}=-1000\left(\frac{2}{5}\right)^{1}=-400$.
The common ratio is $r=\frac{2}{5}$.
The formula for the sum of the first $n$ terms is:
$S_{n}=\frac{a_{1}(1-r^{n})}{1-r}$
$S_{10}=\frac{-400\left[1-\left(\frac{2}{5}\right)^{10}\right]}{1-\frac{2}{5}}\approx-666.597$
