Find the sum of the infinite geometric series. 10 + 5/2 + 5/8 + 5/32 + Write your answer as an integer or a fraction in simplest form. ______

Identify Terms: Identify the first term (a) and the common ratio (r) of the geometric series. The first term a = 10. To find the common ratio r, we divide the second term by the first term. r = (5/2) / 10 = 5/2 * 1/10 = 5/20 = 1/4.Check Convergence: Determine if the series is convergent. A geometric series converges if the absolute value of the common ratio |r| < 1. In this case, |r| = |1/4| = 1/4, which is less than 1. Therefore, the series is convergent.Use Formula: Use the formula for the sum of an infinite geometric series. The sum S of an infinite geometric series is given by S = a / (1 - r), where a is the first term and r is the common ratio.Calculate Sum: Substitute the values of a and r into the formula and calculate the sum. S = 10 / (1 - 1/4) = 10 / (3/4) = 10 * (4/3) = 40/3.

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Find the sum of the infinite geometric series. $\newline$ $10 + \frac{5}{2} + \frac{5}{8} + \frac{5}{32} + \dots$ $\newline$ 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

10 + \frac{5}{2} + \frac{5}{8} + \frac{5}{32} + \dots

\newline

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

\newline

\_\_

Identify Terms: Identify the first term $a$ and the common ratio $r$ of the geometric series. $\newline$ The first term $a = 10$ . $\newline$ To find the common ratio $r$ , we divide the second term by the first term. $\newline$ $r = (\frac{5}{2}) / 10 = \frac{5}{2} \times \frac{1}{10} = \frac{5}{20} = \frac{1}{4}$ .
Check Convergence: Determine if the series is convergent. $\newline$ A geometric series converges if the absolute value of the common ratio \lvert r \rvert < 1. $\newline$ In this case, $\lvert r \rvert = \lvert \frac{1}{4} \rvert = \frac{1}{4}$ , which is less than $1$ . $\newline$ Therefore, the series is convergent.
Use Formula: Use the formula for the sum of an infinite geometric series. $\newline$ The sum $S$ of an infinite geometric series is given by $S = \frac{a}{1 - r}$ , where $a$ is the first term and $r$ is the common ratio.
Calculate Sum: Substitute the values of $a$ and $r$ into the formula and calculate the sum. $S = \frac{10}{(1 - \frac{1}{4})} = \frac{10}{(\frac{3}{4})} = 10 \times (\frac{4}{3}) = \frac{40}{3}.$