Question

Express the recurring decimal 0.1576(bar on 576) as a rational no. Using concept of infinite geometric series.?

Answer 1

Dear Student,

$$\begin{gathered} 0.1\overline{576} \\ =0.1576576576...... \\ =0.1+0.0576+0.0000576+0.0000000576+....... \\ =0.1+\frac{576}{10000}+\frac{576}{10000000}+..... \\ Sincetherepeatingpatternisaninfiniteconverginggeometricserieswhoseratioofsuccessivetermis \\ lessthan1i.e.r=0.001thenwecanusesumofinfinitegeometricseries{S}_{\infty }=\frac{a_1}{1-r} \\ 0.1\overline{576}=0.1+\frac{\frac{576}{10000}}{1-{\displaystyle \frac{1}{1000}}}=\frac{1}{10}+\frac{{\displaystyle \frac{576}{10000}}}{{\displaystyle \frac{999}{1000}}}=\frac{1}{10}+\frac{576}{9990}=\frac{999+576}{9990}=\frac{1575}{9990} \\ Regards. \end{gathered}$$