If S=7 5 +9 5^ 2 +13 5^ 3 +19 5^ 4 + ., then 160 ~S is equal to..… - Vidyadip

Question

If $S=\frac{7}{5}+\frac{9}{5^{2}}+\frac{13}{5^{3}}+\frac{19}{5^{4}}+\ldots .$, then $160 \mathrm{~S}$ is equal to....... .

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Answer

b
$\mathrm{S} =\frac{7}{5}+\frac{9}{5^{2}}+\frac{13}{5^{3}}+\frac{19}{5^{4}}+\ldots$

$\frac{1}{5} \mathrm{~S} =\frac{7}{5^{2}}+\frac{9}{5^{3}}+\frac{13}{5^{4}}+\ldots$

On subtracting

$\frac{4}{5} S=\frac{7}{5}+\frac{2}{5^{2}}+\frac{4}{5^{3}}+\frac{6}{5^{4}}+\ldots$

$S=\frac{7}{4}+\frac{1}{10}\left(1+\frac{2}{5}+\frac{3}{5^{2}}+\ldots\right)$

$S=\frac{7}{4}+\frac{1}{10}\left(1-\frac{1}{5}\right)^{-2}$

$=\frac{7}{4}+\frac{1}{10} \times \frac{25}{16}=\frac{61}{32}$

$\Rightarrow 160 \mathrm{~S}=5 \times 61=305$

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