\(0.2\overline{34}\) = 0.234 34 34 .......
= 0.2 + 0.034 + 0.00034 + 0.0000034 + ...... + ∞
= \(\frac{2}{10}\) + \(\frac{34}{1000}\) + \(\frac{34}{100000}\) + \(\frac{34}{10000000}\) + ..... + ∞
= \(\frac{2}{10}\) + \(\frac{34}{10^3}\)\(\bigg[1+\frac{1}{10^2}+\frac{1}{10^4}+.......\infty\bigg]\)
= \(\frac{2}{10}\) + \(\frac{34}{10^3}\) x \(\bigg(\frac{1}{1-\frac{1}{10^2}}\bigg)\) \(\big(\because\,S_{\infty}=\frac{a}{1-r}.\text{Here}\,a = 1,\,r=\frac{1}{10^2}\big)\)
= \(\frac{2}{10}\) + \(\frac{34}{1000}\) x \(\frac{100}{99}\) = \(\frac{198+34}{990}\) = \(\frac{232}{990}\) = \(\frac{116}{495}.\)
