The first, second and last terms of an A.P. are $\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }$, respectively then the sum of first n terms is 
 

$$\begin{gathered} \mathrm{a}=\mathrm{\alpha } \mathrm{a}+\mathrm{d}=\mathrm{\beta } \\ \mathrm{d}=\mathrm{\beta }-\mathrm{\alpha } \\ \mathrm{\gamma }=a_n=\mathrm{a}+(\mathrm{n}-1)\mathrm{d} \\ \frac{\mathrm{\gamma }-\mathrm{\alpha }}{\mathrm{\beta }-\mathrm{\alpha \beta }}=\mathrm{n}-1 \\ \Rightarrow \mathrm{n}=\frac{\mathrm{\beta }+\mathrm{\gamma }-2\mathrm{\alpha }}{\mathrm{\beta }-\mathrm{\alpha }} \\ {\mathrm{S}}_{\mathrm{n}}=\frac{\mathrm{n}}{2}({\mathrm{a}}_1+{\mathrm{a}}_{\mathrm{n}}) \\ =\frac{\mathrm{\beta }+\mathrm{\gamma }-2\mathrm{\alpha }}{2(\mathrm{\beta }-\mathrm{\alpha })} (\mathrm{\alpha }+\mathrm{\gamma }) \end{gathered}$$