Find the sum of the infinite geometric series. 1 + 3/5 + 9/25 + 27/125 + 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 1. The second term is 3/5, so to find the common ratio (r), we divide the second term by the first term: r = (3/5) / 1 = 3/5.Check Ratio Value: Now we check if the common ratio (r) is less than 1 in absolute value. Since 3/5 is less than 1, we can use the sum formula for an infinite geometric series.Apply Sum Formula: We apply the formula S = a / (1 - r) to find the sum of the series. Substituting the values we have: S = 1 / (1 - 3/5).Calculate Sum: We calculate the sum: S = 1 / (1 - 3/5) = 1 / (2/5) = 1 * (5/2) = 5/2.Final Answer: The sum of the infinite geometric series is 5/2. This is the final answer in its simplest form.

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Find the sum of the infinite geometric series. $\newline$ $1 + \frac{3}{5} + \frac{9}{25} + \frac{27}{125} + \dots$ $\newline$ 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

1 + \frac{3}{5} + \frac{9}{25} + \frac{27}{125} + \dots

\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.
Calculate Common Ratio: The first term $a$ of the series is $1$ . The second term is $\frac{3}{5}$ , so to find the common ratio $r$ , we divide the second term by the first term: $r = \frac{\frac{3}{5}}{1} = \frac{3}{5}$ .
Check Ratio Value: Now we check if the common ratio $r$ is less than $1$ in absolute value. Since $\frac{3}{5}$ is less than $1$ , we can use the sum formula for an infinite geometric series.
Apply Sum Formula: We apply the formula $S = \frac{a}{1 - r}$ to find the sum of the series. Substituting the values we have: $S = \frac{1}{1 - \frac{3}{5}}$ .
Calculate Sum: We calculate the sum: $S = \frac{1}{1 - \frac{3}{5}} = \frac{1}{\frac{2}{5}} = 1 \times \frac{5}{2} = \frac{5}{2}.$
Final Answer: The sum of the infinite geometric series is $\frac{5}{2}$ . This is the final answer in its simplest form.