Answer
$S_{10}=-10$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To find $S_{10}$ in the given sequence
\begin{array}{l}\require{cancel}
8,6,4,...
\end{array}
use the formula for finding 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, the common difference of the given sequence is
\begin{array}{l}\require{cancel}
d=a_2-a_1
\\\\
d=6-8
\\\\
d=-2
.\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 $a_1=8$ 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\cdot8+(10-1)(-2)]
\\\\
S_{10}=5[16+(9)(-2)]
\\\\
S_{10}=5[16-18]
\\\\
S_{10}=5[-2]
\\\\
S_{10}=-10
.\end{array}
