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Question

(C) Let $x = 2.\overline{357} = 2.357357357...$This can be written as $x = 2 + 0.357357357...$Let $y = 0.357357357... = \frac{357}{1000} + \frac{357}{

Vedclass · Accepted Answer

$\frac{2355}{999}$

Vedclass · Answer

(C) Let $x = 2.\overline{357} = 2.357357357...$This can be written as $x = 2 + 0.357357357...$Let $y = 0.357357357... = \frac{357}{1000} + \frac{357}{1000^2} + \frac{357}{1000^3} + ...$This is an infinite geometric series with first term $a = \frac{357}{1000}$ and common ratio $r = \frac{1}{1000}$.The sum $S_{\infty} = \frac{a}{1-r} = \frac{\frac{357}{1000}}{1 - \frac{1}{1000}} = \frac{\frac{357}{1000}}{\frac{999}{1000}} = \frac{357}{999}$.Therefore,$x = 2 + \frac{357}{999} = \frac{2 	imes 999 + 357}{999} = \frac{1998 + 357}{999} = \frac{2355}{999}$.

$2.\overline{357} = $

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