College Algebra, Chapter 9, 9.3, Section 9.3, Problem 24

Determine the first five terms of the sequence $a_n =(-1)^n 2^n$ and determine whether it is geometric. If it is geometric, find the common ratio, and express the $n$th term of the sequence in the standard form $a_n = ar^{n-1}$.

The first five terms are


$
\begin{equation}
\begin{aligned}

a_1 =& (-1)^1 2^1 = -2
\\
\\
a_2 =& (-1)^2 2^2 = 4
\\
\\
a_3 =& (-1)^3 2^3 = -8
\\
\\
a_4 =& (-1)^4 2^4 = 16
\\
\\
a_5 =& (-1)^5 2^5 = -32

\end{aligned}
\end{equation}
$


The sequence $-2,4,-8,16,-32,...$ is a geometric sequence with $a = -2$ and $r = -2$, when $r$ is negative, the terms of the sequence alternate in sign. The $n$th term is $a_n = -2(-2)^{n-1}$