Answer
$90$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To evaluate the given expression, $
\displaystyle\sum_{i=1}^{5} (5i+3)
,$ use the properties of summation.
$\bf{\text{Solution Details:}}$
Using a property of the summation which is given by $\displaystyle\sum_{i=1}^n (a_i+b_i)=\displaystyle\sum_{i=1}^n a_i+\displaystyle\sum_{i=1}^nb_i,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\displaystyle\sum_{i=1}^5 5i+\displaystyle\sum_{i=1}^53
.\end{array}
Using a property of the summation which is given by $\displaystyle\sum_{i=1}^n ca_i=c\displaystyle\sum_{i=1}^n a_i,$ with $c=
2
,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
5\displaystyle\sum_{i=1}^5 i+\displaystyle\sum_{i=1}^53
.\end{array}
Using a property of the summation which is given by $\displaystyle\sum_{i=1}^n i=\dfrac{n(n+1)}{2},$ with $n=
5
,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
5\cdot\dfrac{5(5+1)}{2}+\displaystyle\sum_{i=1}^53
\\\\=
5\cdot\dfrac{5(6)}{2}+\displaystyle\sum_{i=1}^53
\\\\=
5\cdot15+\displaystyle\sum_{i=1}^53
\\\\=
75+\displaystyle\sum_{i=1}^53
.\end{array}
Using a property of the summation which is given by $\displaystyle\sum_{i=1}^n c=nc,$ with $n=
5
,$ and $c=
3
,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
75+5(3)
\\\\=
75+15
\\\\=
90
.\end{array}
