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In a geometric progression with first term $a$ and common ratio $r$, what is the arithmetic mean of first five terms?
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The correct answer is:
$a\left(r^{5}-1\right) /[5(r-1)]$
Let the Geometric progression be a, ar, ar $^{2}, \mathrm{ar}^{3}, \mathrm{ar}^{4}, \ldots$ First five terms of a geometric progression are $\mathrm{a}, \mathrm{ar}, \mathrm{ar}^{2}$, $\mathrm{ar}^{3}, \mathrm{ar}^{4}$
$\begin{array}{ll}\therefore \quad & \text { Mean }=\frac{a+a r+a r^{2}+a r^{3}+a r^{4}}{5} \\ & =\frac{a\left(r^{5}-1\right)}{5(r-1)} \quad\left(\because \text { Sum of G.P }=\frac{a\left(r^{n}-1\right)}{r-1}\right)\end{array}$
$\begin{array}{ll}\therefore \quad & \text { Mean }=\frac{a+a r+a r^{2}+a r^{3}+a r^{4}}{5} \\ & =\frac{a\left(r^{5}-1\right)}{5(r-1)} \quad\left(\because \text { Sum of G.P }=\frac{a\left(r^{n}-1\right)}{r-1}\right)\end{array}$
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