Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 10 - Review Exercises - Page 1125: 50


The repeated decimal $0.\bar{6}$ can be represented in lowest form as $\frac{2}{3}$

Work Step by Step

It can be an infinite geometric sequence as: $\begin{align} & 0.\bar{6}=0.6+0.06+0.006+\ldots \\ & {{a}_{1}}=\frac{6}{10},r=\frac{1}{10} \\ & {{S}_{\infty }}=\frac{\frac{6}{10}}{1-\frac{1}{10}} \end{align}$ Therefore, $\begin{align} & {{S}_{\infty }}=\frac{\frac{6}{10}}{\frac{9}{10}} \\ & {{S}_{\infty }}=\frac{6}{9} \\ & =\frac{2}{3} \end{align}$ Hence, the repeated decimal can be represented in the lowest form as $\frac{2}{3}$.
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