Answer
Geometric;
$-\dfrac{2}{3}(4^n-1)$
Work Step by Step
We are given the sequence:
$\left\{-\dfrac{1}{2}\cdot 4^n\right\}$
Determine the ratio of consecutive terms:
$\dfrac{a_{n+1}}{a_n}=\dfrac{-\dfrac{1}{2}\cdot 4^{n+1}}{-\dfrac{1}{2}\cdot 4^n}=4$
As the ratio of consecutive terms is constant, the sequence is geometric. Its elements are:
$a_1=-\dfrac{1}{2}\cdot 4^1=-\dfrac{1}{2}\cdot 4=-2$
$r=4$
Determine the sum of the first $n$ terms:
$S_n=a_1\dfrac{1-r^n}{1-r}=-2\cdot\dfrac{1-4^n}{1-4}=-\dfrac{2}{3}(4^n-1)$
