Answer
$a_8=\dfrac{85}{3}
\text{ and }
a_n=\dfrac{5}{3} +\dfrac{10}{3}n
$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To find $a_8$ and $a_n$ in the given sequence
\begin{array}{l}\require{cancel}
a_1=5, a_4=15
,\end{array}
use the formula for finding the $n$th term of an arithmetic sequence.
$\bf{\text{Solution Details:}}$
Using $a_n=a_1+(n-1)d$ with $a_1=5,$ $a_4=15,$ and $n=4$ then
\begin{array}{l}\require{cancel}
a_4=a_1+(4-1)d
\\\\
15=5+3d
\\\\
15-5=3d
\\\\
3d=10
\\\\
d=\dfrac{10}{3}
.\end{array}
Using $a_n=a_1+(n-1)d$ with $a_1=5$ and $d=\dfrac{10}{3}$ then
\begin{array}{l}\require{cancel}
a_n=5+(n-1)\left( \dfrac{10}{3} \right)
\\\\
a_n=5+\dfrac{10}{3}n-\dfrac{10}{3}
\\\\
a_n=\left( 5-\dfrac{10}{3} \right) +\dfrac{10}{3}n
\\\\
a_n=\left( \dfrac{15}{3}-\dfrac{10}{3} \right) +\dfrac{10}{3}n
\\\\
a_n=\dfrac{5}{3} +\dfrac{10}{3}n
.\end{array}
With $n=8,$ then
\begin{array}{l}\require{cancel}
a_8=\dfrac{5}{3} +\dfrac{10}{3}\cdot8
\\\\
a_8=\dfrac{5}{3} +\dfrac{80}{3}
\\\\
a_8=\dfrac{85}{3}
.\end{array}
Hence, $
a_8=\dfrac{85}{3}
\text{ and }
a_n=\dfrac{5}{3} +\dfrac{10}{3}n
.$
