Find the sum of the infinite geometric series. -7 - 7/3 - 7/9 - 7/27 + 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. In this series, the first term a is -7, and we can find the common ratio by dividing the second term by the first term, or the third term by the second term, and so on. Let's calculate the common ratio r: r = (-7/3) / (-7) = 1/3.Calculate Common Ratio: Now that we have the first term a = -7 and the common ratio r = 1/3, we can use the formula for the sum of an infinite geometric series: S = a / (1 - r). Let's plug in the values: S = (-7) / (1 - 1/3).Apply Formula: We need to simplify the expression: S = (-7) / (1 - 1/3) = (-7) / (2/3). To divide by a fraction, we multiply by its reciprocal: S = (-7) * (3/2).Simplify Expression: Now, let's perform the multiplication: S = (-7) * (3/2) = -21/2. This is the sum of the infinite geometric series in its simplest form.

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Find the sum of the infinite geometric series. $\newline$ $-7 - \frac{7}{3} - \frac{7}{9} - \frac{7}{27} +$ $\newline$ 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

-7 - \frac{7}{3} - \frac{7}{9} - \frac{7}{27} +

\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. In this series, the first term $a$ is $-7$ , and we can find the common ratio by dividing the second term by the first term, or the third term by the second term, and so on. Let's calculate the common ratio $r$ : $r = \frac{-7/3}{-7} = \frac{1}{3}$ .
Calculate Common Ratio: Now that we have the first term $a = -7$ and the common ratio $r = \frac{1}{3}$ , we can use the formula for the sum of an infinite geometric series: $S = \frac{a}{1 - r}$ . Let's plug in the values: $S = \frac{-7}{1 - \frac{1}{3}}$ .
Apply Formula: We need to simplify the expression: $S = (-7) / (1 - \frac{1}{3}) = (-7) / (\frac{2}{3})$ . $\newline$ To divide by a fraction, we multiply by its reciprocal: $S = (-7) \times (\frac{3}{2})$ .
Simplify Expression: Now, let's perform the multiplication: $S = (-7) \times \left(\frac{3}{2}\right) = -\frac{21}{2}$ . This is the sum of the infinite geometric series in its simplest form.