Answer
$S_{10}=160$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To find $S_{10}$ in the given sequence
\begin{array}{l}\require{cancel}
a_2=9,a_4=13
\end{array}
use the formulas for finding the $n^{th}$ term and the sum of the first $n$ terms of an arithmetic sequence.
$\bf{\text{Solution Details:}}$
Since the fourth term of an arithmetic sequence is 2 terms away from the second term, then,
\begin{array}{l}\require{cancel}
a_4=a_2+2d
.\end{array}
With $a_4=13$ and $a_2=9,$ the equation above becomes
\begin{array}{l}\require{cancel}
13=9+2d
\\\\
13-9=2d
\\\\
4=2d
\\\\
d=\dfrac{4}{2}
\\\\
d=2
.\end{array}
Using $a_n=a_1+(n-1)d$ with $a_2=9$ and $d=2$ then
\begin{array}{l}\require{cancel}
a_n=a_1+(n-1)d
\\\\
a_2=a_1+(2-1)d
\\\\
9=a_1+(2-1)2
\\\\
9=a_1+(1)2
\\\\
9=a_1+2
\\\\
9-2=a_1
\\\\
a_1=7
.\end{array}
Using the formula for the sum of the first $n$ terms of an airthmetic sequence, which is given by $
S_n=\dfrac{n}{2}[2a_1+(n-1)d]
,$ then the sum of the first $n=10$ terms with $a_1=7$ and $d=2$ is
\begin{array}{l}\require{cancel}
S_n=\dfrac{n}{2}[2a_1+(n-1)d]
\\\\
S_{10}=\dfrac{10}{2}[2\cdot7+(10-1)(2)]
\\\\
S_{10}=5[14+(9)(2)]
\\\\
S_{10}=5[14+18]
\\\\
S_{10}=5[32]
\\\\
S_{10}=160
.\end{array}
