Answer
$\displaystyle\sum_{j=1}^{15} (5j-9)=465$
Work Step by Step
$\bf{\text{Solution Outline:}}$
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
.$
