Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 12 - Sequences; Induction; the Binomial Theorem - 12.2 Arithmetic Sequences - 12.2 Assess Your Understanding - Page 814: 54



Work Step by Step

We have to determine the sum: $S=\sum_{n=1}^{80} \left(\dfrac{1}{3}n+\dfrac{1}{2}\right)$ $S$ is the sum of an arithmetic sequence. Determine its first term and the common difference: $a_1=\dfrac{1}{3}(1)+\dfrac{1}{2}=\dfrac{5}{6}$ $d=\left(\dfrac{1}{3}(k+1)+\dfrac{1}{2}\right)-\left(\dfrac{1}{3}k+\dfrac{1}{2}\right)=\dfrac{1}{3}$ The number of terms is 80, so we have to determine the sum of the first $80$ terms. We use the formula: $S_n=\dfrac{n(a_1+a_n)}{2}$ $a_1=\dfrac{5}{6}$ $a_{80}=\dfrac{1}{3}(80)+\dfrac{1}{2}=\dfrac{163}{6}$ $\sum_{n=1}^{80} \left(\dfrac{1}{3}n+\dfrac{1}{2}\right)=\dfrac{80\left(\dfrac{5}{6}+\dfrac{163}{6}\right)}{2}=1120$
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