Cayley Mathematical Olympiad 2021 Q4

Let $S\left(n\right)$ denote the sum of the first $n$ terms of the series $1-2+3-4+5-6+...$For example, $S\left(5\right)=1-2+3-4+5=3$.

Find an expression for $S\left(n\right)$ in the form $an+b$ $n$, when $n$ is an odd number.