College Algebra (11th Edition)

$\displaystyle\sum_{j=1}^{15} (5j-9)=465$
$\bf{\text{Solution Outline:}}$ To evaluate the given expression, $\displaystyle\sum_{j=1}^{15} (5j-9) ,$ 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=5(1)-9 \\\\ a_1=5-9 \\\\ a_1=-4 .\end{array} Substituting $j$ with $15 ,$ then the last term, $a_n,$ is \begin{array}{l}\require{cancel} a_n=5(15)-9 \\\\ a_n=75-9 \\\\ a_n=66 .\end{array} With $j$ going from $1$ to $15 ,$ then there are a total of $15$ terms in the series. Hence, $n= 15 .$ 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_{15}=\dfrac{15}{2}(-4+66) \\\\ S_{15}=\dfrac{15}{2}(62) \\\\ S_{15}=15(31) \\\\ S_{15}=465 .\end{array} Hence, $\displaystyle\sum_{j=1}^{15} (5j-9)=465 .$