Find the fractional value of 0.1232323...... | vedclass

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(C) Let $x = 0.1232323......$ $x = 0.1 + 0.0232323......$ $x = \frac{1}{10} + \frac{23}{1000} + \frac{23}{100000} + ...$ This is a geometric series wi

Vedclass · Accepted Answer

$\frac{61}{495}$

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(C) Let $x = 0.1232323......$ $x = 0.1 + 0.0232323......$ $x = \frac{1}{10} + \frac{23}{1000} + \frac{23}{100000} + ...$ This is a geometric series with first term $a = \frac{23}{1000}$ and common ratio $r = \frac{1}{100}$. Sum of infinite $GP$ is $S = \frac{a}{1-r} = \frac{23/1000}{1 - 1/100} = \frac{23/1000}{99/100} = \frac{23}{990}$. Therefore,$x = \frac{1}{10} + \frac{23}{990} = \frac{99 + 23}{990} = \frac{122}{990} = \frac{61}{495}$.

Find the fractional value of $0.1232323......$

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