Find the sum of the infinite geometric series. -3 - 1 - 1/3 - 1/9 + 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 = -3 To find the common ratio, divide the second term by the first term: r = a_2/a_1 = (-1)/(-3) = 1/3Use sum formula: Use 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 = -3 and r = 1/3.Substitute values: Substitute the values of a_1 and r into the formula to find the sum S. S = (-3)/(1 - (1/3))Simplify denominator: Simplify the denominator of the fraction. 1 - (1/3) = 3/3 - 1/3 = 2/3Calculate sum: Now, calculate the sum S with the simplified denominator. S = (-3)/(2/3)Multiply numerator: Multiply the numerator by the reciprocal of the denominator to find the sum. S = -3 * (3/2) S = -9/2

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Find the sum of the infinite geometric series. $\newline$ $-3 - 1 - \frac{1}{3} - \frac{1}{9} +$ $\newline$ 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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-3 - 1 - \frac{1}{3} - \frac{1}{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. $a_1 = -3$ To find the common ratio, divide the second term by the first term: $r = \frac{a_2}{a_1} = \frac{-1}{-3} = \frac{1}{3}$
Use sum formula: Use 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, $a_1 = -3$ and $r = \frac{1}{3}$ .
Substitute values: Substitute the values of $a_1$ and $r$ into the formula to find the sum $S$ . $\newline$ $S = \frac{-3}{1 - (\frac{1}{3})}$
Simplify denominator: Simplify the denominator of the fraction. $1 - \left(\frac{1}{3}\right) = \frac{3}{3} - \frac{1}{3} = \frac{2}{3}$
Calculate sum: Now, calculate the sum $S$ with the simplified denominator. $S = \frac{-3}{\frac{2}{3}}$
Multiply numerator: Multiply the numerator by the reciprocal of the denominator to find the sum. $\newline$ $S = -3 \times \left(\frac{3}{2}\right)$ $\newline$ $S = -\frac{9}{2}$