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: 78

Answer

$\frac{251}{990}$

Work Step by Step

We express the number as a sum of fractions: $0.25353...=0.2+\frac{53}{1000}+\frac{53}{100,000}+\frac{53}{10,000,000}+...$ We know that this represents an infinite geometric series with $a=0.053$ and $r= \frac{1}{100}$ (added to $0.2$). We know the sum of an infinite geometric series is: $S_{\infty}=\frac{a}{1-r}$ $S_{\infty}=\frac{0.053}{1-\frac{1}{100}}=\frac{53}{990}$ Thus the original number is: $0.2+\frac{53}{990}=\frac{2}{10}+\frac{53}{990}=\frac{2*99+53}{990}=\frac{251}{990}$
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