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 $a_1=2$.
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 $a_1=2$ 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}&=2 \cdot 3^{10-1}\\
&=2\cdot 3^9\\
&=2\cdot 19683\\
&=39366\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{2(1-3^{10})}{1-3}\\\\
&=\frac{2(1-59049)}{-2}\\\\
&=\frac{2(-59048)}{-2}\\\\
&=59048
\end{align*}
