Answer
$S_{10}=125$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To find $S_{10}$ in the given sequence
\begin{array}{l}\require{cancel}
a_3=5, a_4=8
\end{array}
use the formulas for finding the $n^{th}$ term and the sum of the first $n$ terms of an arithmetic sequence.
$\bf{\text{Solution Details:}}$
The common difference, $d,$ of an arithmetic sequence is the difference of a term and the term preceeding it. Hence,
\begin{array}{l}\require{cancel}
d=a_4-a_3
\\\\
d=8-5
\\\\
d=3
.\end{array}
Using $a_n=a_1+(n-1)d$ with $a_3=5$ and $d=3$ then
\begin{array}{l}\require{cancel}
a_n=a_1+(n-1)d
\\\\
a_3=a_1+(3-1)d
\\\\
5=a_1+(2)(3)
\\\\
5=a_1+6
\\\\
5-6=a_1
\\\\
a_1=-1
.\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 $a_1=-1$ and $d=3$ is
\begin{array}{l}\require{cancel}
S_n=\dfrac{n}{2}[2a_1+(n-1)d]
\\\\
S_{10}=\dfrac{10}{2}[2\cdot(-1)+(10-1)(3)]
\\\\
S_{10}=5[-2+(9)(3)]
\\\\
S_{10}=5[-2+27]
\\\\
S_{10}=5[25]
\\\\
S_{10}=125
.\end{array}
