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

Question
Series
asked 2020-12-31
(a) write the repeating decimal as a geometric series, and (b) write its sum as the ratio of two integers. \(0.\overline{81}\)

Answers (1)

2021-01-01
Given decimal: \(0.\overline{81}\)
(a) For the repeating decimal 0.81, we can write
\(0.\overline{81}=0.818181...\)
\(=0.81+0.0081+0.000081+0.00000081+...\)
\(=\frac{81}{10^2}+\frac{81}{10^4}+\frac{81}{10^6}+...\)
\(=\sum_{n=0}^{\infty}\frac{81}{10^2}(\frac{1}{10^2})^n\) (b) Given Series: \(=\sum_{n=0}^{\infty}\frac{81}{10^2}(\frac{1}{10^2})^n\)
Given series is a Geometric series with ratio \(r=\frac{1}{10^2}\) and \(a_1=\frac{81}{10^2}\)
So the sum is \(=\frac{a_1}{(1-r)}\)
\(=\frac{\frac{81}{100}}{(1-\frac{1}{100})}\)
\(=\frac{81}{99}\)
\(=\frac{9}{11}\)
Result
\(\frac{9}{11}\)
0

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