Answer
$S_3=18$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To evaluate the given expression, $
\displaystyle\sum_{i=1}^3 (i+4)
,$ 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+4
\\\\
a_1=5
.\end{array}
Substituting $i$ with $
3
,$ then the last term, $a_n,$ is
\begin{array}{l}\require{cancel}
a_n=3+4
\\\\
a_n=7
.\end{array}
With $i$ going from $
1
$ to $
3
,$ then there are a total of $
3
$ terms in the series. Hence, $n=
3
.$
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_3=\dfrac{3}{2}(5+7)
\\\\
S_3=\dfrac{3}{2}(12)
\\\\
S_3=3(6)
\\\\
S_3=18
.\end{array}
