College Algebra (11th Edition)

Published by Pearson
ISBN 10: 0321671791
ISBN 13: 978-0-32167-179-0

Chapter 7 - Section 7.2 - Arithmetic Sequences and Series - 7.2 Exercises: 55

Answer

$\displaystyle\sum_{j=1}^{10} (2j+3)=140$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To evaluate the given expression, $ \displaystyle\sum_{j=1}^{10} (2j+3) ,$ use the formula for finding the sum of $n$ terms that form an arithmetic sequence. $\bf{\text{Solution Details:}}$ Substituting $j$ with $ 1 ,$ then the first term, $a_1,$ is \begin{array}{l}\require{cancel} a_1=2(1)+3 \\\\ a_1=2+3 \\\\ a_1=5 .\end{array} Substituting $j$ with $ 10 ,$ then the last term, $a_n,$ is \begin{array}{l}\require{cancel} a_n=2(10)+3 \\\\ a_n=20+3 \\\\ a_n=23 .\end{array} With $j$ going from $ 1 $ to $ 10 ,$ then there are a total of $ 10 $ terms in the series. Hence, $n= 10 .$ Using the formula for finding the sum of $n$ terms that form an arithmetic sequence, which is given by $S_n=\dfrac{n}{2}(a_1+a_n),$ then \begin{array}{l}\require{cancel} S_{10}=\dfrac{10}{2}(5+23) \\\\ S_{10}=5(28) \\\\ S_{10}=140 .\end{array} Hence, $ \displaystyle\sum_{j=1}^{10} (2j+3)=140 .$
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