Find the sum of the infinite geometric series. 8 + 32/5 + 128/25 + 512/125 + Write your answer as an integer or a fraction in simplest form. ______

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 = 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): r = (32/5) / 8 = 32 / (5 * 8) = 32 / 40 = 4 / 5 We can check this by dividing the third term by the second term: r = (128/25) / (32/5) = (128/25) * (5/32) = 4 / 5 The common ratio is consistent, so r = 4/5.Simplify Expression: Now that we have the first term (a = 8) and the common ratio (r = 4/5), we can use the formula for the sum of an infinite geometric series to find the sum: S = a / (1 - r) = 8 / (1 - 4/5)Final Sum: Simplify the expression: S = 8 / (1 - 4/5) = 8 / (5/5 - 4/5) = 8 / (1/5) = 8 * (5/1) = 40The sum of the infinite geometric series is 40.

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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.

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

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

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\_\_

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 $40$ .