Let 517+55(17)^2+555(17)^3+....∞=S ⇒ 10S17=50(17)^2+550(17)^3+......∞ ⇒ 10S17+517+5(17)^2+5(17)^3+.....∞ =517+55(17)^2+555(17)^3 ...

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Standard Formulae - 1

517+55172+555...

Question

$\frac{5}{17} + \frac{55}{(17)^{2}} + \frac{555}{(17)^{3}} + . . . . \infty$ .

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Solution

Let $\frac{5}{17} + \frac{55}{{(17)}^{2}} + \frac{555}{{(17)}^{3}} + . . . . \infty = S$
$\Rightarrow$ $\frac{10 S}{17} = \frac{50}{{(17)}^{2}} + \frac{550}{{(17)}^{3}} + . . . . . . \infty$
$\Rightarrow$ $\frac{10 S}{17} + \frac{5}{17} + \frac{5}{{(17)}^{2}} + \frac{5}{{(17)}^{3}} + . . . . . \infty$
$= \frac{5}{17} + \frac{55}{{(17)}^{2}} + \frac{555}{{(17)}^{3}} + . . . . . . \infty$
$\Rightarrow$ $\frac{10 S}{17} + 5. \frac{\frac{1}{17}}{1 - \frac{1}{17}} = S \Rightarrow \frac{5}{16} = \frac{7 S}{17} \Rightarrow S = \frac{85}{112}$

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