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Question Number 139 by shaleen last updated on 25/Jan/15

$$\mathit{e}\mathit{x}\mathit{p}\mathit{r}\mathit{e}\mathit{s}\mathit{s}\mathit{t}\mathit{h}\mathit{e}\mathit{f}\mathit{o}\mathit{l}\mathit{o}\mathit{w}\mathit{i}\mathit{n}\mathit{g}\mathit{i}\mathit{n}\mathit{t}\mathit{h}\mathit{e}0.6\left(\mathit{b}\mathit{a}\mathit{r}\right)=$$$$\mathit{f}\mathit{o}\mathit{r}\mathit{m}\mathit{o}\mathit{f}\mathit{p}/\mathit{q};wherepandq$$$$areintegersandqisnot=0$$$$0.\overline{6}$$

Answered by vkulkarni last updated on 10/Dec/14

$$0.\bar{6}=0.6666666....$$$$=\frac{6}{10}+\frac{6}{100}+\frac{6}{1000}+\frac{6}{{10}^4}+....$$$$\mathrm{This}\mathrm{is}\mathrm{an}\mathrm{infinite}\mathrm{G}.\mathrm{P}.\mathrm{with}\mathrm{first}$$$$\mathrm{term},a=\frac{6}{10}\mathrm{and}r=\frac{1}{10},\mathrm{hence}$$$$0.\stackrel{-}{6}=\frac{a}{1-r}=\frac{\frac{6}{10}}{1-\frac{1}{10}}=\frac{6}{9}=\frac{2}{3}$$

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