Find the sum of the infinite geometric series. 9 + 36/5 + 144/25 + 576/125 + Write your answer as an integer or a fraction in simplest form. ______

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

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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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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{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 $45$ .