Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 12 - Section 12.3 - Geometric Sequences - 12.3 Exercises - Page 865: 81

Answer

$\displaystyle \frac{112}{999}$

Work Step by Step

$ 0.\overline{112}=0.112112112\ldots$ = $=0.112+0.000112+0.000000112+...$ $=\displaystyle \frac{112}{1000}+\frac{112}{1,000,000}+\frac{112}{1,000,000,000}+\cdots$ ... an infinite geometric series, with $a=\displaystyle \frac{112}{1000}$ and $r=\displaystyle \frac{1}{1000}$. $|\displaystyle \frac{1}{1000}| < 1$, so the sum exists, $S=\displaystyle \frac{a}{1-r}$ $0.\displaystyle \overline{112}=\frac{\frac{112}{1000}}{1-\frac{1}{1000}}=\frac{\frac{112}{1000}}{\frac{999}{1000}}$ $=\displaystyle \frac{112}{999}$
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