Sum of n terms of A.G.P.

Let $S_n=a+(a+d)r+(a+2d)r^2+\cdots+[a+(n-1)d]r^{n-1}$. Multiplying by $r$: $rS_n=ar+(a+d)r^2+(a+2d)r^3+\cdots+[a+(n-1)d]r^n$. Subtracting, the differences between successive bracketed terms all become $d$, leaving a genuine G.P. of $(n-1)$ terms in $d$, plus the leftover boundary terms: $S_n(1-r)=a+dr+dr^2+\cdots+dr^{n-1}-[a+(n-1)d]r^n=a+\dfrac{dr(1-r^{n-1})}{1-r}-[a+(n-1)d]r^n$. Dividing by $(1-r)$:

The sum to infinity of an A.G.P. (when $|r|<1$, so the last term above vanishes) is $S_\infty=\dfrac{a}{1-r}+\dfrac{dr}{(1-r)^2}$. …