Question

Find the value of $0.1\overline{23}$ regarding it as a geometric series

Answer 1

$\cdot \dot{12}\dot{3}=\cdot 123232323.$
$=\frac{1}{10}+\frac{23}{1000}+\frac{23}{100000}+....$
$=\frac{1}{10}+23[\frac{1}{{10}^3}+\frac{1}{{10}^5}+\frac{1}{{10}^7}+.....\infty ]$
$=\frac{1}{10}+23\cdot \frac{\left(1/{10}^3\right)}{1-\left(1/{10}^2\right)}=\frac{1}{10}+\frac{23}{990}$ $\because S=\frac{a}{1-r}$
$=\frac{99+23}{990}=\frac{122}{990}=\frac{61}{495}$