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  4. The sum displaystyle∑ k =1∞ (2 k +2/3 k ) equal to

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Q. The sum $\displaystyle\sum_{ k =1}^{\infty} \frac{2^{ k +2}}{3^{ k }}$ equal to

Sequences and Series

Solution:

$S =4\displaystyle \sum_{ k =1}^{\infty}\left(\frac{2}{3}\right)^{ k }=4\left[\frac{2}{3}+\left(\frac{2}{3}\right)^2+\left(\frac{2}{3}\right)^3+\ldots \ldots ..\right]=4\left[\frac{2 / 3}{1-(2 / 3)}\right]=8$