To identify the first term [tex]\(a_1\)[/tex] and the common ratio [tex]\(r\)[/tex] for the given geometric series, follow these steps:
1. Identify the first term [tex]\(a_1\)[/tex]:
- The first term of the series is the first number in the sequence provided.
- For the series [tex]\(-4, 8, -16, 32, \ldots\)[/tex], the first term [tex]\(a_1\)[/tex] is clearly [tex]\(-4\)[/tex].
2. Identify the common ratio [tex]\(r\)[/tex]:
- The common ratio [tex]\(r\)[/tex] in a geometric series is the factor by which each term is multiplied to get the next term.
- To find [tex]\(r\)[/tex], divide the second term by the first term:
[tex]\[
r = \frac{\text{second term}}{\text{first term}}
\][/tex]
- For the series [tex]\(-4, 8, -16, 32, \ldots\)[/tex]:
[tex]\[
r = \frac{8}{-4} = -2
\][/tex]
Thus, the first term [tex]\(a_1\)[/tex] is [tex]\(-4\)[/tex], and the common ratio [tex]\(r\)[/tex] is [tex]\(-2\)[/tex].
So we have:
[tex]\[
\begin{array}{l}
a_1 = -4 \\
r = -2
\end{array}
\][/tex]
