Answer
$2032$
Work Step by Step
The sum of the first $n$ terms can be computed as: $S_n=\dfrac{a_1(1-r^{n})}{1-r}$
where, $a_1$ is first term and $r$ is the common ratio, $r$ and can be computed as the quotient of a term and the term preceding it. In a geometric series the ratio between consecutive terms is constant.
We can see from the given series that $a_4=2^4=16$ and $r=2$.
The given series is the sum of $7$ terms of a geometric series with $2^k$ as the kth term , that is, $a_k$. So, we will consider $n=7$
Plug $7$ for $n$ and $16$ for $a_4$ and $2$ for $r$ in the equation above.
So, the sum of the first $7$ terms can be computed as:
$$S_7=\dfrac{16[1-(2)^{7}]}{1-2}=\dfrac{16[1-128]}{-1} =2032$$
