Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 4 - Integration - 4.2 Exercises - Page 263: 22

Answer

$S(n)=\displaystyle \frac{7n+15}{2n}$ $S(10)=\displaystyle \frac{17}{4}=4.25$ $S(100)=3.575$ $S(1000)=3.5075$ $S(10,000)=3.50075$

Work Step by Step

$S(n)= \displaystyle \sum_{j=1}^{n}\frac{7j+4}{n^{2}}=$ ... the $n^{2}$ is constant,...$\displaystyle \sum_{i=1}^{n}ka_{i}=k\sum_{i=1}^{n}a_{i}$ =$\displaystyle \frac{1}{n^{2}}\sum_{j=\mathrm{I}}^{n}(7j+4)$ ... property: $ \displaystyle \sum_{i=1}^{n}(a_{i}\pm b_{i})=\sum_{i=1}^{n}a_{i}\pm\sum_{i=1}^{n}b_{i}$ =$\displaystyle \frac{1}{n^{2}}[\sum_{j=\mathrm{I}}^{n}(7j)+\sum_{j=\mathrm{I}}^{n}4]$ ... Th4.2.2. $\displaystyle \sum_{i=1}^{n}i=\frac{n(n+1)}{2}$, Th4.2.1. $\displaystyle \sum_{i=1}^{n}c=cn$ $=\displaystyle \frac{1}{n^{2}}[7\cdot\frac{n(n+1)}{2}+4n]$ $=\displaystyle \frac{7n^{2}+7n}{2n^{2}}+\frac{4n}{n^{2}}=\frac{n(7n+7)}{2n^{2}}+\frac{4}{n}$ $=\displaystyle \frac{7n+7+8}{2n}$ $S(n)=\displaystyle \frac{7n+15}{2n}$ $S(10)=\displaystyle \frac{70+15}{20}=4.25$ $S(100)=\displaystyle \frac{700+15}{200}=3.575$ $S(1000)=\displaystyle \frac{7000+15}{2000}=3.5075$ $S(10,000)=\displaystyle \frac{70,000+15}{20,000}=3.50075$
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