Find the sum of the infinite geometric series. 8 + 6 + 9/2 + 27/8 + 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 the first number in the series, which is 8. To find the common ratio r, we divide the second term by the first term, the third term by the second term, and so on, to ensure they all give the same ratio. r = 6 / 8 = 9/2 / 6 = 27/8 / 9/2 r = 3/4Calculate Common Ratio: Now that we have the first term a = 8 and the common ratio r = 3/4, which is less than 1, we can use the formula for the sum of an infinite geometric series: S = a / (1 - r)Use Formula for Sum: Let's plug the values into the formula to find the sum: S = 8 / (1 - 3/4) S = 8 / (1/4) S = 8 * (4/1) S = 32Plug in Values: We have calculated the sum of the series to be 32. This is the final answer, and we should check it once more to ensure there are no errors.

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Find the sum of the infinite geometric series. $\newline$ $8 + 6 + \frac{9}{2} + \frac{27}{8} + \dots$ $\newline$ 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

8 + 6 + \frac{9}{2} + \frac{27}{8} + \dots

\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. The first term $a$ is the first number in the series, which is $8$ . To find the common ratio $r$ , we divide the second term by the first term, the third term by the second term, and so on, to ensure they all give the same ratio. $r = \frac{6}{8} = \frac{9}{2} / 6 = \frac{27}{8} / \frac{9}{2}$ $r = \frac{3}{4}$
Calculate Common Ratio: Now that we have the first term $a = 8$ and the common ratio $r = \frac{3}{4}$ , which is less than $1$ , we can use the formula for the sum of an infinite geometric series: $\newline$ $S = \frac{a}{(1 - r)}$
Use Formula for Sum: Let's plug the values into the formula to find the sum: $\newline$ $S = \frac{8}{1 - \frac{3}{4}}$ $\newline$ $S = \frac{8}{\frac{1}{4}}$ $\newline$ $S = 8 \times \frac{4}{1}$ $\newline$ $S = 32$
Plug in Values: We have calculated the sum of the series to be $32$ . This is the final answer, and we should check it once more to ensure there are no errors.