Answer
The sequence is geometric and the sum of its first ten terms is $-59,048$.
Work Step by Step
The terms have a common ratio of $-3$ so the sequence is geometric with $r=-3$ and $a_1=4$.
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 $a_1=4$ and $r=-3$.
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}&=4 \cdot (-3)^{10-1}\\
&=4\cdot (-3)^9\\
&=4(-19683)\\
&=-78,732\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{4(1-(-3)^{10})}{1-(-3)}\\\\
&=\frac{4(1-59049}{1+3}\\\\
&=\frac{4(-59048)}{4}\\\\
&=-59048
\end{align*}
