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NSW Y12 Maths - Advanced Sequences and Series Recurring Decimals

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Recurring Decimals Theory

A repeating decimal can be changed into a fraction by using the sum to infinity of a geometric series viz \(S_n=\dfrac{a}{1-r}\).\\ \textbf{Example}\\ %11994 Using the limiting sum of a GP \(0.\dot 1\dot 2\dot 5\) as a fraction is?\\ \textbf{Solution}\\ $\begin{aligned} 0.125 &=0.125125125 \ldots \\ &=\frac{125}{1000}+\frac{125}{1000000}+\frac{125}{1000000000} \\ a=\frac{125}{1000} &=\frac{1}{8}, \ r =\frac{1}{1000} \\ S_{\infty} &=\frac{a}{1-r}=\frac{\frac{1}{8}}{1-\frac{1}{1000}}\\&=\frac{1}{8} \div \frac{999}{1000} \\ &=\frac{125}{999} \end{aligned}$\\

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Videos

Videos relating to Recurring Decimals.

  • Recurring Decimals - Video- Recurring Decimals

  • Recurring Decimals - Video - Geometric Series Sum to infinity writing recurring decimals as fractions

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