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Find the sum of the infinite geometric series. $\newline$ $8 + \frac{32}{5} + \frac{128}{25} + \frac{512}{125} +$ $\newline$ 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

8 + \frac{32}{5} + \frac{128}{25} + \frac{512}{125} +

\newline

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

\newline

\_\_

Identify Terms and 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 $|r| < 1$ .
Calculate Common Ratio: The first term $a$ of the series is the first number in the sequence, which is $8$ . We need to find the common ratio $r$ by dividing the second term by the first term, the third term by the second term, and so on, to ensure it is consistent.
Use Formula for Sum: Calculating the common ratio $r$ : $\newline$ $r = \frac{32}{5} / 8 = \frac{32}{5 \times 8} = \frac{32}{40} = \frac{4}{5}$ $\newline$ We can check this by dividing the third term by the second term: $\newline$ $r = \frac{128}{25} / \frac{32}{5} = \frac{128}{25} \times \frac{5}{32} = \frac{4}{5}$ $\newline$ The common ratio is consistent, so $r = \frac{4}{5}$ .
Simplify Expression: Now that we have the first term $a = 8$ and the common ratio $r = \frac{4}{5}$ , we can use the formula for the sum of an infinite geometric series to find the sum: $\newline$ $S = \frac{a}{1 - r} = \frac{8}{1 - \frac{4}{5}}$
Final Sum: Simplify the expression: $S = \frac{8}{1 - \frac{4}{5}} = \frac{8}{\frac{5}{5} - \frac{4}{5}} = \frac{8}{\frac{1}{5}} = 8 \times \frac{5}{1} = 40$
Final Sum: Simplify the expression: $\newline$ $S = \frac{8}{1 - \frac{4}{5}} = \frac{8}{\frac{5}{5} - \frac{4}{5}} = \frac{8}{\frac{1}{5}} = 8 \times \frac{5}{1} = 40$ The sum of the infinite geometric series is $40$ .

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