Solve a=s/n-(n-1)alpha/2

$$a = \frac { s } { n } - \frac { ( n - 1 ) \alpha } { 2 }$$

Solve for n (complex solution)

$$\left\{\begin{matrix}n=\frac{-\sqrt{8s\alpha -4a\alpha +4a^{2}+\alpha ^{2}}+\alpha -2a}{2\alpha }\text{, }&\left(\alpha \neq 0\text{ and }a\neq \frac{\alpha }{2}\text{ and }arg(2a-\alpha )<\pi \right)\text{ or }\left(\alpha \neq 0\text{ and }s\neq 0\right)\\n=\frac{\sqrt{8s\alpha -4a\alpha +4a^{2}+\alpha ^{2}}+\alpha -2a}{2\alpha }\text{, }&\left(\alpha \neq 0\text{ and }arg(2a-\alpha )\geq \pi \text{ and }a\neq \frac{\alpha }{2}\right)\text{ or }\left(\alpha \neq 0\text{ and }s\neq 0\right)\\n=\frac{s}{a}\text{, }&s\neq 0\text{ and }a\neq 0\text{ and }\alpha =0\\n\neq 0\text{, }&\alpha =0\text{ and }a=0\text{ and }s=0\end{matrix}\right.$$

Solve for a

$$a=-\frac{n\alpha }{2}+\frac{\alpha }{2}+\frac{s}{n}$$

$$n\neq 0$$