Cayley Mathematical Olympiad 2021 Q5

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

It can be show that when:$\qquad n$ is even $S\left(n\right)=-\frac{n}{2}$ $\qquad n$ is odd $S\left(n\right)=\frac{n+1}{2}$

Find the value of $S\left(a\right)+S\left(b\right)+S\left(a+b\right)$ when $a$ and $b$ are both odd.