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: First, we need to identify the first term (a) and the common ratio (r) of the geometric series. The first term a is -8. To find the common ratio r, we divide the second term by the first term: r = (-32/5) / (-8) = 4/5.Calculate common ratio: Now that we have the first term and the common ratio, we can use the formula for the sum of an infinite geometric series, which is S = a / (1 - r), provided that |r| < 1. In this case, |4/5| < 1, so we can use the formula.Use formula for sum: Let's plug the values into the formula: S = (-8) / (1 - (4/5)).Calculate denominator: Now we calculate the denominator: 1 - (4/5) = 5/5 - 4/5 = 1/5.Calculate sum: Next, we calculate the sum: S = (-8) / (1/5). To divide by a fraction, we multiply by its reciprocal: S = (-8) * (5/1) = -40.

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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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Precalculus

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: First, we need to identify the first term $a$ and the common ratio $r$ of the geometric series. $\newline$ The first term $a$ is $-8$ . $\newline$ To find the common ratio $r$ , we divide the second term by the first term: $r = (-32/5) / (-8) = 4/5$ .
Calculate common ratio: Now that we have the first term and the common ratio, we can use the formula for the sum of an infinite geometric series, which is $S = \frac{a}{1 - r}$ , provided that |r| < 1. In this case, |\frac{4}{5}| < 1, so we can use the formula.
Use formula for sum: Let's plug the values into the formula: $S = \frac{-8}{1 - \left(\frac{4}{5}\right)}$ .
Calculate denominator: Now we calculate the denominator: $1 - \left(\frac{4}{5}\right) = \frac{5}{5} - \frac{4}{5} = \frac{1}{5}$ .
Calculate sum: Next, we calculate the sum: $S = (-8) / (\frac{1}{5})$ . $\newline$ To divide by a fraction, we multiply by its reciprocal: $S = (-8) \times (\frac{5}{1}) = -40$ .