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Find the sum of the infinite geometric series. $\newline$ $5 + \frac{5}{2} + \frac{5}{4} + \frac{5}{8} + \dots$ $\newline$ Write your answer as an integer or a fraction in simplest form. $\newline$ $\_\_$

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Find the value of an infinite geometric series

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Q. Find the sum of the infinite geometric series.

\newline

5 + \frac{5}{2} + \frac{5}{4} + \frac{5}{8} + \dots

\newline

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

\newline

\_\_

Identify first term and common 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 we multiply to get from one term to the next. $\newline$ In this series, the first term $a$ is $5$ , and the common ratio $r$ is $\frac{1}{2}$ , because each term is half of the previous term.
Use formula for sum: The sum $S$ of an infinite geometric series can be found using the formula $S = \frac{a}{1 - r}$ , where $a$ is the first term and $r$ is the common ratio, but only if $|r| < 1$ . Since the common ratio here is $\frac{1}{2}$ , which is less than $1$ , we can use this formula.
Plug in values: Now we plug the values of $a$ and $r$ into the formula to find the sum of the series: $\newline$ $S = \frac{5}{1 - \frac{1}{2}}$
Calculate denominator: We calculate the denominator of the fraction: $1 - \frac{1}{2} = \frac{1}{2}$
Divide first term: Now we divide the first term by the result we just found: $S = \frac{5}{(\frac{1}{2})}$
Multiply by reciprocal: To divide by a fraction, we multiply by its reciprocal. So we multiply $5$ by $\frac{2}{1}$ (which is the reciprocal of $\frac{1}{2}$ ): $S = 5 \times \left(\frac{2}{1}\right)$
Perform multiplication: We perform the multiplication: $S = 10$

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