Answer
$a_n=\frac{11}{2}-\frac{3}{2}n$, $S_{10}=-\frac{55}{2}$
Work Step by Step
Given the sequence $4,\frac{5}{2},1,-\frac{1}{2},-2,...$, we have $\frac{5}{2}-4=-\frac{3}{2}, 1-\frac{5}{2}=-\frac{3}{2},...$ and it can be identified as an arithmetic sequence. We have $a_n=4+(n-1)(-\frac{3}{2})=\frac{11}{2}-\frac{3}{2}n$ and $S_{10}=\frac{10}{2}(4+\frac{11}{2}-\frac{3}{2}(10))=-\frac{55}{2}$
