Simplify: sum_(n=1)^(oo)12((2)/(5))^(n-1)

Rewrite series in geometric format: Simplify the series formula to a geometric series format. The series can be rewritten as 12 * sum_(n=1)^(oo)((2/5)^(n-1)). This is a geometric series with the first term a = (2/5)^0 = 1 and common ratio r = 2/5.Apply sum formula for geometric series: Apply the formula for the sum of an infinite geometric series. The sum S of an infinite geometric series where |r| < 1 is given by S = a / (1 - r). Here, a = 1 and r = 2/5, so S = 1 / (1 - 2/5) = 1 / (3/5) = 5/3.Calculate final sum: Multiply the result by 12 to find the final sum of the original series. The final sum is 12 * (5/3) = 20.

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Simplify: sum_(n=1)^(oo)12((2)/(5))^(n-1)

Simplify: $\sum_{n=1}^{\infty}12\left(\frac{2}{5}\right)^{n-1}$

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Algebra 2

Sum of finite series starts from 1

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Q. Simplify:

\sum_{n=1}^{\infty}12\left(\frac{2}{5}\right)^{n-1}

Rewrite series in geometric format: Simplify the series formula to a geometric series format. $\newline$ The series can be rewritten as $12 \times \sum_{n=1}^{\infty}\left(\frac{2}{5}\right)^{n-1}$ . $\newline$ This is a geometric series with the first term $a = \left(\frac{2}{5}\right)^0 = 1$ and common ratio $r = \frac{2}{5}$ .
Apply sum formula for geometric series: Apply the formula for the sum of an infinite geometric series. $\newline$ The sum $S$ of an infinite geometric series where |r| < 1 is given by $S = \frac{a}{1 - r}$ . $\newline$ Here, $a = 1$ and $r = \frac{2}{5}$ , so $S = \frac{1}{1 - \frac{2}{5}} = \frac{1}{\frac{3}{5}} = \frac{5}{3}$ .
Calculate final sum: Multiply the result by $12$ to find the final sum of the original series. $\newline$ The final sum is $12 \times \left(\frac{5}{3}\right) = 20$ .