Find the sum of the infinite geometric series. 3 + 3/2 + 3/4 + 3/8 + 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 first term is the first number in the series, and the common ratio is the factor by which each term is multiplied to get the next term. In this series, the first term (a) is 3, and the common ratio (r) can be found by dividing the second term by the first term, or the third term by the second term, and so on. Calculating the common ratio: r = (3/2) / 3 = 1/2Calculate Common Ratio: The sum S of an infinite geometric series with |r| < 1 is given by the formula S = a / (1 - r), where a is the first term and r is the common ratio. Plugging in the values we have: S = 3 / (1 - 1/2)Calculate Sum: Now we perform the calculation for the sum S. S = 3 / (1 - 1/2) = 3 / (1/2) = 3 * (2/1) = 6 The sum of the infinite geometric series is 6.

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Find the sum of the infinite geometric series. $\newline$ $3 + \frac{3}{2} + \frac{3}{4} + \frac{3}{8} + \dots$ $\newline$ 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

3 + \frac{3}{2} + \frac{3}{4} + \frac{3}{8} + \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 first term is the first number in the series, and the common ratio is the factor by which each term is multiplied to get the next term. $\newline$ In this series, the first term $a$ is $3$ , and the common ratio $r$ can be found by dividing the second term by the first term, or the third term by the second term, and so on. $\newline$ Calculating the common ratio: $r = (\frac{3}{2}) / 3 = \frac{1}{2}$
Calculate Common Ratio: The sum $S$ of an infinite geometric series with |r| < 1 is given by the formula $S = \frac{a}{1 - r}$ , where $a$ is the first term and $r$ is the common ratio. $\newline$ Plugging in the values we have: $S = \frac{3}{1 - \frac{1}{2}}$
Calculate Sum: Now we perform the calculation for the sum $S$ . $\newline$ $S = \frac{3}{1 - \frac{1}{2}} = \frac{3}{\frac{1}{2}} = 3 \times \frac{2}{1} = 6$ $\newline$ The sum of the infinite geometric series is $6$ .