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Find the sum of the infinite geometric series. $\newline$ $1 + \frac{2}{5} + \frac{4}{25} + \frac{8}{125} + \dots$ $\newline$ 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

Full solution

Q. Find the sum of the infinite geometric series.

\newline

1 + \frac{2}{5} + \frac{4}{25} + \frac{8}{125} + \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. $\newline$ The first term $a$ is $1$ . $\newline$ The common ratio $r$ is the factor by which we multiply each term to get the next term. In this case, we multiply by $\frac{2}{5}$ to get from one term to the next. $\newline$ So, $r = \frac{2}{5}$ .
Apply Geometric Series Formula: Next, we use the formula for the sum of an infinite geometric series, which is $S = \frac{a}{1 - r}$ , but only if the absolute value of $r$ is less than $1$ . Since $|r| = \left|\frac{2}{5}\right| = 0.4$ , which is less than $1$ , we can use the formula.
Calculate Sum: Now, we plug the values of $a$ and $r$ into the formula to find the sum $S$ . $\newline$ $S = \frac{1}{1 - \frac{2}{5}}$
Calculate Denominator: We calculate the denominator of the fraction: $1 - \frac{2}{5} = \frac{5}{5} - \frac{2}{5} = \frac{3}{5}$
Find Final Sum: Now, we find the sum $S$ by dividing the first term by the result we just found: $S = \frac{1}{(\frac{3}{5})}$
Multiply by Reciprocal: To divide by a fraction, we multiply by its reciprocal. So, we multiply $1$ by $\frac{5}{3}$ : $S = 1 \times \left(\frac{5}{3}\right) = \frac{5}{3}$

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