Question

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

Answer 1

Let the rational number S.

$\because S=0.125+0.000125+0.000000125+...\infty$

$\Rightarrow S=\frac{125}{{10}^3}+\frac{125}{{10}^6}+\frac{125}{{10}^9}+....$

$\Rightarrow S=\frac{125}{{10}^3}(1+\frac{1}{{10}^3}+\frac{1}{{10}^6}+....)$

$S=\frac{125}{{10}^3}\left(\frac{1}{1-\frac{1}{1000}}\right)$

$S=\frac{125}{1000}(\frac{1000}{999}=\frac{125}{999})$