Find the sum of the infinite geometric series. 4 + 8/5 + 16/25 + 32/125 + 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 first term in this series is 4, and each subsequent term is multiplied by 8/5 divided by the previous term, which is 2/5. So, the common ratio r is 2/5.Use sum formula: The sum of an infinite geometric series can be found using the formula S = 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 = 2/5.Plug in values: Now we plug the values into the formula to calculate the sum: S = 4 / (1 - 2/5).Simplify denominator: Simplify the denominator: 1 - 2/5 = 5/5 - 2/5 = 3/5.Divide by denominator: Now, divide the first term by the simplified denominator: S = 4 / (3/5).Multiply by reciprocal: To divide by a fraction, we multiply by its reciprocal. So, S = 4 * (5/3).Calculate final sum: Multiply the numerators and denominators: S = (4 * 5) / 3 = 20 / 3.

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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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Precalculus

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: $S = (4 \times 5) / 3 = 20 / 3$ .