# Express the recurring decimal 0.125125125 ...

Question:

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

Solution:

Let the rational number $S$ be $0 . \overline{125}$.

$\because \mathrm{S}=0 . \overline{125}=0.125+0.000125+0.000000125+0.000000000125+\ldots \infty$

$\Rightarrow \mathrm{S}=0.125\left[1+10^{-3}+10^{-6}+10^{-9}+\ldots \infty\right]$

Clearly, $\mathrm{S}$ is a geometric series with the first term, $a$, being 1 and the common ratio, $r$, being $10^{-3}$.

$\therefore S=\frac{1}{(1-r)}$

$\Rightarrow \mathrm{S}=0.125\left[\frac{1}{1-10^{-3}}\right]$

$\Rightarrow \mathrm{S}=\frac{125}{999}$