What is the value of [tex]\( a_1 \)[/tex] of the geometric series?

[tex]\[
\sum_{n=1}^{\infty} 12\left(-\frac{1}{9}\right)^{n-1}
\][/tex]

A. [tex]\(-\frac{12}{9}\)[/tex]

B. [tex]\(-\frac{1}{9}\)[/tex]

C. 1

D. 12



Answer :

To determine the value of [tex]\( a_1 \)[/tex] for the given geometric series

[tex]\[ \sum_{n=1}^{\infty} 12\left(-\frac{1}{9}\right)^{n-1}, \][/tex]

let's break down the series into its components.

A geometric series is of the form:

[tex]\[ \sum_{n=1}^{\infty} a_1 \cdot r^{n-1}, \][/tex]

where:
- [tex]\( a_1 \)[/tex] is the first term,
- [tex]\( r \)[/tex] is the common ratio.

In the given series:

[tex]\[ \sum_{n=1}^{\infty} 12\left(-\frac{1}{9}\right)^{n-1}, \][/tex]

we can identify the first term [tex]\( a_1 \)[/tex] and the common ratio [tex]\( r \)[/tex].

Here, the first term [tex]\( a_1 \)[/tex] is the coefficient outside the exponentiated term, which is 12.

So,

[tex]\[ a_1 = 12. \][/tex]

Thus, the value of [tex]\( a_1 \)[/tex] is:

12