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Find the sum of the infinite geometric series. $\newline$ $-8 - \frac{24}{5} - \frac{72}{25} - \frac{216}{125} +$ $\newline$ Write your answer as an integer or a fraction in simplest form. $\newline$ ______

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Find the value of an infinite geometric series

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Q. Find the sum of the infinite geometric series.

\newline

-8 - \frac{24}{5} - \frac{72}{25} - \frac{216}{125} +

\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 $-8$ . $\newline$ To find the common ratio $r$ , we divide the second term by the first term. $\newline$ $r = \frac{-24}{5} / (-8) = \frac{3}{5}$
Check Convergence: Check if the absolute value of the common ratio is less than $1$ to ensure the series converges. $\newline$ $|\frac{3}{5}| = \frac{3}{5}$ , which is less than $1$ , so the series converges.
Use Sum Formula: Use the formula for the sum of an infinite geometric series, which is $S = \frac{a}{1 - r}$ , where $S$ is the sum, $a$ is the first term, and $r$ is the common ratio. $\newline$ $S = \frac{-8}{1 - (\frac{3}{5})}$
Calculate Sum: Calculate the sum using the formula. $\newline$ $S = \frac{-8}{1 - \left(\frac{3}{5}\right)}$ $\newline$ $S = \frac{-8}{\frac{2}{5}}$ $\newline$ $S = -8 \times \left(\frac{5}{2}\right)$ $\newline$ $S = \frac{-40}{2}$ $\newline$ $S = -20$

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