Answer
$-1925$
Work Step by Step
There are $100$ terms, and they form an arithmetic sequence with
$d=a_{n+1}-a_{n}=6-\displaystyle \frac{1}{2}(n+1)-(6-\frac{1}{2}n)$
$=6-\displaystyle \frac{1}{2}n-\frac{1}{2}-6+\frac{1}{2}n$
$=-\displaystyle \frac{1}{2}$
$a_{1}=6-\displaystyle \frac{1}{2}(1)=\frac{11}{2},$
$a_{100}=6-\displaystyle \frac{1}{2}(100)=-44$
Sum of the First $n$ Terms of an arithmetic sequence:
$S_{n}=\displaystyle \frac{n}{2}\left(a_{1}+a_{n}\right)$
$S_{100}=\displaystyle \frac{100}{2}\left(\frac{11}{2}+(-44)\right)$
$=50\displaystyle \left(-\frac{77}{2}\right)=-25\cdot 77=-1925$
