## College Algebra (11th Edition)

$\displaystyle\sum_{j=1}^{10} (2j+3)=140$
$\bf{\text{Solution Outline:}}$ 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, $\displaystyle\sum_{j=1}^{10} (2j+3)=140 .$