Answer
$\frac{10457}{4950}$
Work Step by Step
We express the number as a sum of fractions:
$2.112525.. =2.11+ \frac{25}{10,000}+\frac{25}{1,000,000}+\frac{25}{100,000,000}+...$
We know that this represents an infinite geometric series with $a=\frac{25}{10000}$ and $r= \frac{1}{100}$ (added to $2.11$).
We know the sum of an infinite geometric series is:
$S_{\infty}=\frac{a}{1-r}$
$S_{\infty}=\frac{\frac{25}{10000}}{1-\frac{1}{100}}=\frac{25}{9900}$
We add this to $2.11$:
$2.11+\frac{25}{9900}=\frac{211}{100}+\frac{25}{9900}=\frac{211*99+25}{9900}=\frac{20914}{9900}=\frac{10457}{4950}$
