Find the sum of the infinite geometric series. 9 + 27/4 + 81/16 + 243/64 + Write your answer as an integer or a fraction in simplest form. ______

Identify first term and common ratio: First, we need to identify the first term (a) and the common ratio (r) of the geometric series. The first term a is the first number in the series, which is 9. The common ratio r is found by dividing the second term by the first term, which is (27/4) / 9.Calculate common ratio: Now, let's calculate the common ratio r. r = (27/4) / 9 r = (27/4) * (1/9) r = 27 / 36 r = 3 / 4Use formula for infinite series: Since we have an infinite geometric series, we can use the formula for the sum of an infinite geometric series, which is S = a / (1 - r), provided that the absolute value of r is less than 1. Here, |r| = |3/4|, which is less than 1, so we can use the formula.Apply formula to find sum: Let's apply the formula to find the sum S. S = a / (1 - r) S = 9 / (1 - 3/4) S = 9 / (4/4 - 3/4) S = 9 / (1/4) S = 9 * (4/1) S = 36

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Find the sum of the infinite geometric series. $\newline$ $9 + \frac{27}{4} + \frac{81}{16} + \frac{243}{64} +$ $\newline$ 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{27}{4} + \frac{81}{16} + \frac{243}{64} +

\newline

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

\newline

______

Identify first term and common ratio: First, we need to identify the first term $a$ and the common ratio $r$ of the geometric series. $\newline$ The first term $a$ is the first number in the series, which is $9$ . $\newline$ The common ratio $r$ is found by dividing the second term by the first term, which is $\frac{27}{4} / 9$ .
Calculate common ratio: Now, let's calculate the common ratio $r$ . $r = \frac{27}{4} / 9$ $r = \frac{27}{4} \times \frac{1}{9}$ $r = \frac{27}{36}$ $r = \frac{3}{4}$
Use formula for infinite series: Since we have an infinite geometric series, we can use the formula for the sum of an infinite geometric series, which is $S = \frac{a}{(1 - r)}$ , provided that the absolute value of $r$ is less than $1$ . Here, $|r| = |\frac{3}{4}|$ , which is less than $1$ , so we can use the formula.
Apply formula to find sum: Let's apply the formula to find the sum $S$ . $S = \frac{a}{1 - r}$ $S = \frac{9}{1 - \frac{3}{4}}$ $S = \frac{9}{\frac{4}{4} - \frac{3}{4}}$ $S = \frac{9}{\frac{1}{4}}$ $S = 9 \times \frac{4}{1}$ $S = 36$