Find the sum of the infinite geometric series. -6 - 2 - 2/3 - 2/9 + Write your answer as an integer or a fraction in simplest form. ______

Identify terms and ratio: Identify the first term (a_1) and the common ratio (r) of the geometric series. a_1 = -6 To find r, we use the formula r = a_2/a_1. r = (-2)/(-6) = 1/3Apply sum formula: Apply the formula for the sum of an infinite geometric series, which is S = a_1/(1 - r), where S is the sum, a_1 is the first term, and r is the common ratio. Here, a_1 = -6 and r = 1/3. Calculate the sum using these values. S = (-6) / (1 - (1/3))Simplify expression: Simplify the expression to find the sum. S = (-6) / (2/3) To divide by a fraction, multiply by its reciprocal. S = (-6) * (3/2)Perform multiplication: Perform the multiplication to get the final answer. S = -18/2 S = -9

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Find the sum of the infinite geometric series. $\newline$ $-6 - 2 - \frac{2}{3} - \frac{2}{9} +$ $\newline$ Write your answer as an integer or a fraction in simplest form. $\newline$ ______

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Algebra 2

Find the value of an infinite geometric series

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Q. Find the sum of the infinite geometric series.

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-6 - 2 - \frac{2}{3} - \frac{2}{9} +

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

\newline

______

Identify terms and ratio: Identify the first term $a_1$ and the common ratio $r$ of the geometric series. $a_1 = -6$ To find $r$ , we use the formula $r = \frac{a_2}{a_1}$ . $r = \frac{-2}{-6} = \frac{1}{3}$
Apply sum formula: Apply the formula for the sum of an infinite geometric series, which is $S = \frac{a_1}{1 - r}$ , where $S$ is the sum, $a_1$ is the first term, and $r$ is the common ratio. $\newline$ Here, $a_1 = -6$ and $r = \frac{1}{3}$ . $\newline$ Calculate the sum using these values. $\newline$ $S = \frac{-6}{1 - (\frac{1}{3})}$
Simplify expression: Simplify the expression to find the sum. $\newline$ $S = \frac{-6}{\frac{2}{3}}$ $\newline$ To divide by a fraction, multiply by its reciprocal. $\newline$ $S = (-6) \times \frac{3}{2}$
Perform multiplication: Perform the multiplication to get the final answer. $\newline$ $S = \frac{-18}{2}$ $\newline$ $S = -9$