Find the sum of the finite geometric series. sum_(n = 1)^6 2 * 2^(n – 1) ____

Identify terms and ratio: Identify the first term and the common ratio of the geometric series. The first term a_1 is given when n = 1, which is 2 * 2^(1 – 1) = 2 * 2^0 = 2 * 1 = 2. The common ratio r is 2 because each term is multiplied by 2 to get the next term.Use series formula: Use the formula for the sum of a finite geometric series. The sum S_n of the first n terms of a geometric series is given by the formula S_n = a_1 * (1 - r^n) / (1 - r), where a_1 is the first term, r is the common ratio, and n is the number of terms.Plug in values: Plug the values into the formula. Here, a_1 = 2, r = 2, and n = 6. S_6 = 2 * (1 - 2^6) / (1 - 2)Calculate sum: Calculate the sum. S_6 = 2 * (1 - 64) / (1 - 2) S_6 = 2 * (-63) / (-1) S_6 = 2 * 63 S_6 = 126

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Find the sum of the finite geometric series. $\newline$ $\sum_{n = 1}^{6} 2 \cdot 2^{(n – 1)}$ $\newline$ ______

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Algebra 2

Find the sum of a finite geometric series

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Q. Find the sum of the finite geometric series.

\newline

\sum_{n = 1}^{6} 2 \cdot 2^{(n – 1)}

\newline

______

Identify terms and ratio: Identify the first term and the common ratio of the geometric series. $\newline$ The first term $a_1$ is given when $n = 1$ , which is $2 \times 2^{1 – 1} = 2 \times 2^0 = 2 \times 1 = 2$ . $\newline$ The common ratio $r$ is $2$ because each term is multiplied by $2$ to get the next term.
Use series formula: Use the formula for the sum of a finite geometric series. $\newline$ The sum $S_n$ of the first $n$ terms of a geometric series is given by the formula $S_n = a_1 \times (1 - r^n) / (1 - r)$ , where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.
Plug in values: Plug the values into the formula. $\newline$ Here, $a_1 = 2$ , $r = 2$ , and $n = 6$ . $\newline$ $S_6 = 2 \times (1 - 2^6) / (1 - 2)$
Calculate sum: Calculate the sum. $\newline$ $S_6 = 2 \times (1 - 64) / (1 - 2)$ $\newline$ $S_6 = 2 \times (-63) / (-1)$ $\newline$ $S_6 = 2 \times 63$ $\newline$ $S_6 = 126$