Answer
$S_5=-25$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To evaluate the given expression, $
\displaystyle\sum_{i=1}^5 (i-8)
,$ use the formula for finding the sum of $n$ terms that form an arithmetic sequence.
$\bf{\text{Solution Details:}}$
Substituting $i$ with $
1
,$ then the first term, $a_1,$ is
\begin{array}{l}\require{cancel}
a_1=1-8
\\\\
a_1=-7
.\end{array}
Substituting $i$ with $
5
,$ then the last term, $a_n,$ is
\begin{array}{l}\require{cancel}
a_n=5-8
\\\\
a_n=-3
.\end{array}
With $i$ going from $
1
$ to $
5
,$ then there are a total of $
5
$ terms in the series. Hence, $n=
5
.$
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_5=\dfrac{5}{2}(-7+(-3))
\\\\
S_5=\dfrac{5}{2}(-7-3)
\\\\
S_5=\dfrac{5}{2}(-10)
\\\\
S_5=5(-5)
\\\\
S_5=-25
.\end{array}
