Answer
The sequence is geometric and the sum of its first ten terms is $3069$.
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}\\\\
&=\frac{3(-1023)}{-1}\\\\
&=3069
\end{align*}
