Answer
The sequence is geometric and the sum of its first ten terms is $-1023$.
Work Step by Step
The terms have a common ratio of $-2$ so the sequence is geometric with $r=-2$ and $a_1=3$.
The sum $S_n$ of the first $n$ terms of a geometric sequence is given by the formula:
$$S_n=\frac{a_1(1-r^n)}{1-r}$$
The given sequence has $a_1=3$ and $r=-2$.
However, the value of $a_{10}$ (the tenth term) is not yet known.
Solve for $a_{10}$ using the formula $a_n=a_1\cdot r^{n-1}$ to obtain:
\begin{align*}
a_{10}&=3 \cdot (-2)^{10-1}\\
&=3\cdot (-2)^9\\
&=3\cdot (-512)\\
&=-1536\end{align*}
Use the formula for the sum of the first $n$ terms to find the sum of the first $10$ terms (use $n=10$) to obtain:
\begin{align*}
S_{10}&=\frac{a_1(1-r^n)}{1-r}\\\\
&=\frac{3(1-(-2)^{10})}{1-(-2)}\\\\
&=\frac{3(1-1024)}{1+2}\\\\
&=\frac{3(-1023)}{3}\\\\
&=-1023
\end{align*}
