Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

Published by Pearson
ISBN 10: 0-32193-104-1
ISBN 13: 978-0-32193-104-7

Chapter 11 - Sequences; Induction; the Binomial Theorem - Chapter Test - Page 860: 12

Answer

$\dfrac{1024}{5}$

Work Step by Step

An infinite geometric series converges if and only if $|r|\lt1$, and then its sum is equal to $\dfrac{a_1}{1-r}$ where $r$ is known as the common ratio of the quotient of two consecutive terms and $a_1$ is the first term. Now, the given series has the common ratio: $r=\dfrac{a_2}{a_1}=\dfrac{-64}{256}=-\dfrac{1}{4}$ Thus, $\left|-\dfrac{1}{4}\right|=\dfrac{1}{4}\lt1$ shows that the series converges. We are given that $a_1=256$ Thus, the sum is equal to: $\dfrac{a_1}{1-r} =\dfrac{256}{1-(-\frac{1}{4})}=(256)(\dfrac{4}{5})=\dfrac{1024}{5}$
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