Find the sum of the infinite geometric series. 2 + 3/2 + 9/8 + 27/32 + 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 the given series, the first term is 2 and each subsequent term is multiplied by 3/4 (for example, 3/2 is 3/4 of 2, 9/8 is 3/4 of 3/2, and so on). Therefore, a = 2 and r = 3/4.Apply Sum Formula: Now we apply the formula for the sum of an infinite geometric series: S = a / (1 - r). Substitute the values of a and r into the formula: S = 2 / (1 - 3/4).Simplify Expression: Next, we simplify the expression: S = 2 / (1 - 3/4) = 2 / (4/4 - 3/4) = 2 / (1/4).Calculate Sum: Finally, we calculate the sum: S = 2 / (1/4) = 2 * (4/1) = 8. The sum of the infinite geometric series is 8.

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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.

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2 + \frac{3}{2} + \frac{9}{8} + \frac{27}{32} + \dots

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Write your answer as an integer or a fraction in simplest form.

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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 the given series, the first term is $2$ and each subsequent term is multiplied by $\frac{3}{4}$ (for example, $\frac{3}{2}$ is $\frac{3}{4}$ of $2$ , $\frac{9}{8}$ is $\frac{3}{4}$ of $\frac{3}{2}$ , and so on). Therefore, $r$ $2$ and $r$ $3$ .
Apply Sum Formula: Now we apply the formula for the sum of an infinite geometric series: $S = \frac{a}{1 - r}$ . Substitute the values of $a$ and $r$ into the formula: $S = \frac{2}{1 - \frac{3}{4}}$ .
Simplify Expression: Next, we simplify the expression: $S = \frac{2}{1 - \frac{3}{4}} = \frac{2}{\frac{4}{4} - \frac{3}{4}} = \frac{2}{\frac{1}{4}}$ .
Calculate Sum: Finally, we calculate the sum: $S = \frac{2}{\left(\frac{1}{4}\right)} = 2 \times \left(\frac{4}{1}\right) = 8$ . The sum of the infinite geometric series is $8$ .