Find the sum of the infinite geometric series. -9 - 18/5 - 36/25 - 72/125 + Write your answer as an integer or a fraction in simplest form. ______

Identify Terms and Formula: To find the sum of an infinite geometric series, we need to identify the first term (a) and the common ratio (r) of the series. The formula for the sum of an infinite geometric series is S = a / (1 - r), where |r| < 1. The first term of the series is -9, and we can find the common ratio by dividing the second term by the first term. r = (-18/5) / (-9) = (-18/5) * (-1/9) = 18 / 45 = 2 / 5Calculate Common Ratio: Now that we have the first term a = -9 and the common ratio r = 2/5, we can use the formula for the sum of an infinite geometric series: S = a / (1 - r) S = -9 / (1 - 2/5)Apply Formula for Sum: We need to simplify the denominator of the fraction: 1 - 2/5 = 5/5 - 2/5 = 3/5 So the sum S becomes: S = -9 / (3/5)Simplify Denominator: To divide by a fraction, we multiply by its reciprocal: S = -9 * (5/3) S = -45 / 3 S = -15

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Find the sum of the infinite geometric series. $\newline$ $-9 - \frac{18}{5} - \frac{36}{25} - \frac{72}{125} +$ $\newline$ 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

-9 - \frac{18}{5} - \frac{36}{25} - \frac{72}{125} +

\newline

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

\newline

______

Identify Terms and Formula: To find the sum of an infinite geometric series, we need to identify the first term $a$ and the common ratio $r$ of the series. The formula for the sum of an infinite geometric series is $S = \frac{a}{1 - r}$ , where |r| < 1. The first term of the series is $-9$ , and we can find the common ratio by dividing the second term by the first term. $r = \frac{-18/5}{-9} = \frac{-18/5}{-1/9} = \frac{18}{45} = \frac{2}{5}$
Calculate Common Ratio: Now that we have the first term $a = -9$ and the common ratio $r = \frac{2}{5}$ , we can use the formula for the sum of an infinite geometric series: $\newline$ $S = \frac{a}{1 - r}$ $\newline$ $S = \frac{-9}{1 - \frac{2}{5}}$
Apply Formula for Sum: We need to simplify the denominator of the fraction: $\newline$ $1 - \frac{2}{5} = \frac{5}{5} - \frac{2}{5} = \frac{3}{5}$ $\newline$ So the sum $S$ becomes: $\newline$ $S = -\frac{9}{\left(\frac{3}{5}\right)}$
Simplify Denominator: To divide by a fraction, we multiply by its reciprocal: $\newline$ $S = -9 \times \left(\frac{5}{3}\right)$ $\newline$ $S = -\frac{45}{3}$ $\newline$ $S = -15$