College Algebra 7th Edition

Published by Brooks Cole
ISBN 10: 1305115546
ISBN 13: 978-1-30511-554-5

Chapter 8, Sequences and Series - Section 8.3 - Geometric Sequences - 8.3 Exercises: 81

Answer

$\frac{112}{999}$

Work Step by Step

We express the number as a sum of fractions: $0.\overline{112}=0.11211211\ldots=\frac{112}{1000}+\frac{112}{1,000,000}+\frac{112}{1,000,000,000}+...$ We know that this represents an infinite geometric series with $a=\frac{112}{1000}$ and $r= \frac{1}{1000}$. We know the sum of an infinite geometric series is: $S_{\infty}=\frac{a}{1-r}$ $S_{\infty}=\frac{\frac{112}{1000}}{1-\frac{1}{1000}}=\frac{112}{999}$
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