Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

Published by Pearson
ISBN 10: 0-32184-874-8
ISBN 13: 978-0-32184-874-1

Chapter 14 - Sequences, Series, and the Binomial Theorem - 14.3 Geometric Sequences and Series - 14.3 Exercise Set - Page 912: 57

Answer

$\frac{343}{99}$

Work Step by Step

$3.4646\ldots $ This can be written as, $3.4646\ldots =3+0.4646\ldots $ And, $0.464646\ldots =0.46+0.0046+0.000046+\cdots $ This is an infinite geometric series: So, the value of $\left| r \right|$ is, $\begin{align} & \left| r \right|=\left| \frac{0.0046}{0.46} \right| \\ & =\left| 0.01 \right| \\ & =0.01 \end{align}$ Find the limit of the infinite geometric series by using the formula ${{S}_{\infty }}=\frac{{{a}_{1}}}{1-r}$. $\begin{align} & {{S}_{\infty }}=\frac{{{a}_{1}}}{1-r} \\ & =\frac{0.46}{1-0.01} \\ & =\frac{0.46}{0.99} \\ & =\frac{46}{99} \end{align}$ The fraction notation of the decimal number $0.4646\ldots $ is $\frac{46}{99}$. And the fraction notation of the decimal number $3.4646\ldots $ is: $3+\frac{46}{99}=\frac{343}{99}$ Thus, the fraction notation of the decimal number $3.4646\ldots $ is $\frac{343}{99}$.
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