# (a) write the repeating decimal as a geometric series, and (b) write its sum as the ratio of two integers. 0.overline{81}

(a) write the repeating decimal as a geometric series, and (b) write its sum as the ratio of two integers. $0.\stackrel{―}{81}$
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Nathanael Webber
Given decimal: $0.\stackrel{―}{81}$
(a) For the repeating decimal 0.81, we can write
$0.\stackrel{―}{81}=0.818181...$
$=0.81+0.0081+0.000081+0.00000081+...$
$=\frac{81}{{10}^{2}}+\frac{81}{{10}^{4}}+\frac{81}{{10}^{6}}+...$
$=\sum _{n=0}^{\mathrm{\infty }}\frac{81}{{10}^{2}}\left(\frac{1}{{10}^{2}}{\right)}^{n}$ (b) Given Series: $=\sum _{n=0}^{\mathrm{\infty }}\frac{81}{{10}^{2}}\left(\frac{1}{{10}^{2}}{\right)}^{n}$
Given series is a Geometric series with ratio $r=\frac{1}{{10}^{2}}$ and ${a}_{1}=\frac{81}{{10}^{2}}$
So the sum is $=\frac{{a}_{1}}{\left(1-r\right)}$
$=\frac{\frac{81}{100}}{\left(1-\frac{1}{100}\right)}$
$=\frac{81}{99}$
$=\frac{9}{11}$
Result
$\frac{9}{11}$