Express the recurring decimal 0.125125125 ... as a rational numbe… - Vidyadip

Question

Express the recurring decimal 0.125125125 ... as a rational number.

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Answer

$0.125125125 \ \dots=0.\overline{125}$
$=0.125+0.000125+0.000000125+\ \dots$
$=\frac{125}{10}^3\Big(1+\frac{1}{10^3}+\frac{1}{10^6}+\ \dots\Big)$
$=\frac{125}{10^3}\Bigg(\frac{1}{1-\frac{1}{1000}}\Bigg)$
$=\frac{125}{1000}\Big(\frac{1000}{999}\Big)$
$0.125125125\ \dots=\frac{125}{999}$

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