Find the sum of the infinite geometric series. 10 + 10/3 + 10/9 + 10/27 + Write your answer as an integer or a fraction in simplest form. ______

Identify first term and common ratio: First, we need to identify the first term (a) and the common ratio (r) of the geometric series. The first term a is the first number in the series, which is 10. The common ratio r is the factor that each term is multiplied by to get the next term. To find it, we can divide the second term by the first term: (10/3) / 10 = 1/3.Calculate common ratio: Next, we check if the common ratio's absolute value is less than 1, which is a necessary condition for the sum of an infinite geometric series to exist. Since |r| = |1/3| = 1/3, which is less than 1, the series converges and we can find the sum.Check convergence condition: Now, we use the formula for the sum of an infinite geometric series, which is S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio. Substitute the values of a and r into the formula: S = 10 / (1 - 1/3).Use formula for sum: We calculate the sum by simplifying the expression. S = 10 / (1 - 1/3) = 10 / (3/3 - 1/3) = 10 / (2/3) = 10 * (3/2) = 15.

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Find the sum of the infinite geometric series. $\newline$ $10 + \frac{10}{3} + \frac{10}{9} + \frac{10}{27} + \dots$ $\newline$ 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{10}{3} + \frac{10}{9} + \frac{10}{27} + \dots

\newline

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

\newline

\_\_

Identify first term and common 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 the first number in the series, which is $10$ . $\newline$ The common ratio $r$ is the factor that each term is multiplied by to get the next term. To find it, we can divide the second term by the first term: $(\frac{10}{3}) / 10 = \frac{1}{3}$ .
Calculate common ratio: Next, we check if the common ratio's absolute value is less than $1$ , which is a necessary condition for the sum of an infinite geometric series to exist. $\newline$ Since $|r| = |\frac{1}{3}| = \frac{1}{3}$ , which is less than $1$ , the series converges and we can find the sum.
Check convergence condition: Now, we use the formula for the sum of an infinite geometric series, which is $S = \frac{a}{1 - r}$ , where $S$ is the sum, $a$ is the first term, and $r$ is the common ratio. $\newline$ Substitute the values of $a$ and $r$ into the formula: $S = \frac{10}{1 - \frac{1}{3}}$ .
Use formula for sum: We calculate the sum by simplifying the expression. $\newline$ $S = \frac{10}{1 - \frac{1}{3}} = \frac{10}{\frac{3}{3} - \frac{1}{3}} = \frac{10}{\frac{2}{3}} = 10 \times \frac{3}{2} = 15$ .