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Find the sum of the infinite geometric series. $\newline$ $4 + \frac{8}{5} + \frac{16}{25} + \frac{32}{125} +$ $\newline$ 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

4 + \frac{8}{5} + \frac{16}{25} + \frac{32}{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 first term in this series is $4$ , and each subsequent term is multiplied by $\frac{8}{5}$ divided by the previous term, which is $\frac{2}{5}$ . So, the common ratio $r$ is $\frac{2}{5}$ .
Use sum formula: The sum of an infinite geometric series can be found using the formula $S = \frac{a}{1 - r}$ , where $S$ is the sum of the series, $a$ is the first term, and $r$ is the common ratio. We have $a = 4$ and $r = \frac{2}{5}$ .
Plug in values: Now we plug the values into the formula to calculate the sum: $S = \frac{4}{1 - \frac{2}{5}}$ .
Simplify denominator: Simplify the denominator: $1 - \frac{2}{5} = \frac{5}{5} - \frac{2}{5} = \frac{3}{5}$ .
Divide by denominator: Now, divide the first term by the simplified denominator: $S = \frac{4}{\left(\frac{3}{5}\right)}$ .
Multiply by reciprocal: To divide by a fraction, we multiply by its reciprocal. So, $S = 4 \times \left(\frac{5}{3}\right)$ .
Calculate final sum: Multiply the numerators and denominators: $S = (4 \times 5) / 3 = 20 / 3$ .

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