Assuming you want the sum of the first [tex]n[/tex] terms:
[tex]S_n=a_1+a_2+\cdots+a_{n-1}+a_n[/tex]
[tex]S_n=a_1+ra_1+\cdots+r^{n-2}a_1+r^{n-1}a_1[/tex]
[tex]rS_n=ra_1+r^2a_1+\cdots+r^{n-1}a_1+r^na_1[/tex]
[tex]\implies S_n-rS_n=a_1-r^na_1\iff(1-r)S_n=(1-r^n)a_1\implies S_n=\dfrac{1-r^n}{1-r}a_1[/tex]
You're given that [tex]a_1=-1[/tex] and [tex]r=-2[/tex], so
[tex]S_n=\dfrac{1-(-2)^n}{1-(-2)}(-1)=-\dfrac{1-(-2)^n}3[/tex]
The fact that you're given [tex]a_{10}[/tex] makes me think you're supposed to find [tex]S_{10}[/tex], which is just a matter of setting [tex]n=10[/tex] in the formula above.
[tex]S_{10}=-\dfrac{1-(-2)^{10}}3=-\dfrac{1-1024}3=341[/tex]
