Find the sum of the infinite geometric series. 5 + 15/4 + 45/16 + 135/64 + Write your answer as an integer or a fraction in simplest form. ______

Identify first term and common ratio: 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. In the given series, the first term is 5, and we can find the common ratio by dividing the second term by the first term.Calculate common ratio: Calculate the common ratio (r) by dividing the second term (15/4) by the first term (5). r = (15/4) / 5 r = (15/4) * (1/5) r = 15/20 r = 3/4Use formula for sum: Now that we have the first term (a = 5) and the common ratio (r = 3/4), we can use the formula for the sum of an infinite geometric series. S = a / (1 - r) S = 5 / (1 - 3/4)Calculate sum: Calculate the sum (S) by evaluating the expression. S = 5 / (1 - 3/4) S = 5 / (4/4 - 3/4) S = 5 / (1/4) S = 5 * (4/1) S = 20

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Find the sum of the infinite geometric series. $\newline$ $5 + \frac{15}{4} + \frac{45}{16} + \frac{135}{64} + \dots$ $\newline$ 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

5 + \frac{15}{4} + \frac{45}{16} + \frac{135}{64} + \dots

\newline

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

\newline

\_\_

Identify first term and common ratio: 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. In the given series, the first term is $5$ , and we can find the common ratio by dividing the second term by the first term.
Calculate common ratio: Calculate the common ratio $r$ by dividing the second term $\frac{15}{4}$ by the first term $5$ . $r = \frac{\frac{15}{4}}{5}$ $r = \frac{15}{4} \times \frac{1}{5}$ $r = \frac{15}{20}$ $r = \frac{3}{4}$
Use formula for sum: Now that we have the first term $a = 5$ and the common ratio $r = \frac{3}{4}$ , we can use the formula for the sum of an infinite geometric series. $S = \frac{a}{1 - r}$ $S = \frac{5}{1 - \frac{3}{4}}$
Calculate sum: Calculate the sum $S$ by evaluating the expression. $\newline$ $S = \frac{5}{1 - \frac{3}{4}}$ $\newline$ $S = \frac{5}{\frac{4}{4} - \frac{3}{4}}$ $\newline$ $S = \frac{5}{\frac{1}{4}}$ $\newline$ $S = 5 \times \frac{4}{1}$ $\newline$ $S = 20$