Find the sum of the infinite geometric series. 7 + 21/4 + 63/16 + 189/64 + 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 7. To find the common ratio (r), we divide the second term by the first term. r = (21/4) / 7 r = 21 / (4 * 7) r = 21 / 28 r = 3/4Calculate common ratio: Now that we have the first term (a = 7) and the common ratio (r = 3/4), we can use the formula for the sum of an infinite geometric series, which is S = a / (1 - r), provided that |r| < 1. In this case, |3/4| < 1, so we can use the formula. S = 7 / (1 - 3/4)Use formula for sum: Next, we calculate the sum using the formula. S = 7 / (1 - 3/4) S = 7 / (4/4 - 3/4) S = 7 / (1/4) S = 7 * (4/1) S = 28

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Find the sum of the infinite geometric series. $\newline$ $7 + \frac{21}{4} + \frac{63}{16} + \frac{189}{64} + \dots$ $\newline$ 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

7 + \frac{21}{4} + \frac{63}{16} + \frac{189}{64} + \dots

\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 $7$ . $\newline$ To find the common ratio $r$ , we divide the second term by the first term. $\newline$ $r = \frac{21}{4} / 7$ $\newline$ $r = \frac{21}{4 \times 7}$ $\newline$ $r = \frac{21}{28}$ $\newline$ $r = \frac{3}{4}$
Calculate common ratio: Now that we have the first term $a = 7$ and the common ratio $r = \frac{3}{4}$ , we can use the formula for the sum of an infinite geometric series, which is $S = \frac{a}{1 - r}$ , provided that |r| < 1. $\newline$ In this case, |\frac{3}{4}| < 1, so we can use the formula. $\newline$ $S = \frac{7}{1 - \frac{3}{4}}$
Use formula for sum: Next, we calculate the sum using the formula. $\newline$ $S = \frac{7}{1 - \frac{3}{4}}$ $\newline$ $S = \frac{7}{\frac{4}{4} - \frac{3}{4}}$ $\newline$ $S = \frac{7}{\frac{1}{4}}$ $\newline$ $S = 7 \times \frac{4}{1}$ $\newline$ $S = 28$