Answer
$341/512$.
Work Step by Step
We know that $2^9=512$. Thus, we have $-1/512$ as the the tenth term of the sequence.
The sum of a geometric sequence until $n$ can be obtained by the formula
$S_n=a_1(\frac{1-r^n}{1-r})$
where $a_1$ is the first term and $r$ is the common ratio.
Hence, here the sum is:
$S_{10}=1(\frac{1-(-0.5)^{10}}{1-(-0.5)})=341/512$
