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Find the sum of the infinite geometric series. $\newline$ $9 + \frac{36}{5} + \frac{144}{25} + \frac{576}{125} +$ $\newline$ 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

9 + \frac{36}{5} + \frac{144}{25} + \frac{576}{125} +

\newline

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

\newline

______

Identify Terms: First, we need to identify the first term $a$ and the common ratio $r$ of the geometric series. $\newline$ The first term is $9$ . $\newline$ To find the common ratio, we divide the second term by the first term: $\frac{36}{5} \div 9 = \frac{36}{5} \times \frac{1}{9} = \frac{4}{5}$ . $\newline$ So, the common ratio $r$ is $\frac{4}{5}$ .
Calculate Common Ratio: Now, we can 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. $\newline$ We only apply this formula if the absolute value of $r$ is less than $1$ , which is true in this case since $|\frac{4}{5}| < 1$ .
Apply Sum Formula: Let's plug the values into the formula: $\newline$ $S = \frac{9}{1 - \frac{4}{5}}$ $\newline$ $S = \frac{9}{\frac{5}{5} - \frac{4}{5}}$ $\newline$ $S = \frac{9}{\frac{1}{5}}$ $\newline$ To divide by a fraction, we multiply by its reciprocal. $\newline$ $S = 9 \times \frac{5}{1}$ $\newline$ $S = 45$
Calculate Sum: The sum of the infinite geometric series is $45$ .

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