Find the sum of the infinite geometric series. -6 - 9/2 - 27/8 - 81/32 + Write your answer as an integer or a fraction in simplest form. ______

Identify Terms: Identify the first term (a) and the common ratio (r) of the geometric series. The first term a is -6. To find the common ratio r, we divide the second term by the first term. r = (-9/2) / (-6) = 3/4Convergence Check: Determine if the series is convergent. A geometric series converges if the absolute value of the common ratio |r| is less than 1. In this case, |3/4| = 0.75, which is less than 1. Therefore, the series is convergent.Infinite Series Formula: Use the formula for the sum of an infinite geometric series. The sum S of an infinite geometric series is given by the formula S = a / (1 - r), where a is the first term and r is the common ratio.Calculate Sum: Substitute the values of a and r into the formula to find the sum. S = (-6) / (1 - (3/4)) S = (-6) / (1/4) S = (-6) * (4/1) S = -24

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Find the sum of the infinite geometric series. $\newline$ $-6 - \frac{9}{2} - \frac{27}{8} - \frac{81}{32} +$ $\newline$ Write your answer as an integer or a fraction in simplest form. $\newline$ ______

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Precalculus

Find the value of an infinite geometric series

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Q. Find the sum of the infinite geometric series.

\newline

-6 - \frac{9}{2} - \frac{27}{8} - \frac{81}{32} +

\newline

Write your answer as an integer or a fraction in simplest form.

\newline

______

Identify Terms: Identify the first term $a$ and the common ratio $r$ of the geometric series. $\newline$ The first term $a$ is $-6$ . $\newline$ To find the common ratio $r$ , we divide the second term by the first term. $\newline$ $r = \frac{-9/2}{-6} = \frac{3}{4}$
Convergence Check: Determine if the series is convergent. $\newline$ A geometric series converges if the absolute value of the common ratio $\lvert r \rvert$ is less than $1$ . $\newline$ In this case, $\lvert \frac{3}{4} \rvert = 0.75$ , which is less than $1$ . $\newline$ Therefore, the series is convergent.
Infinite Series Formula: Use the formula for the sum of an infinite geometric series. $\newline$ The sum $S$ of an infinite geometric series is given by the formula $S = \frac{a}{1 - r}$ , where $a$ is the first term and $r$ is the common ratio.
Calculate Sum: Substitute the values of $a$ and $r$ into the formula to find the sum. $\newline$ $S = \frac{-6}{1 - \left(\frac{3}{4}\right)}$ $\newline$ $S = \frac{-6}{\frac{1}{4}}$ $\newline$ $S = (-6) \times \left(\frac{4}{1}\right)$ $\newline$ $S = -24$