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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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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. $\newline$ $S = \frac{7}{1 - \frac{3}{4}}$ $\newline$ $S = \frac{7}{\frac{4}{4} - \frac{3}{4}}$ $\newline$ $S = \frac{7}{\frac{1}{4}}$ $\newline$ $S = 7 \times \frac{4}{1}$ $\newline$ $S = 28$

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