Sum of Arithmetic, Geometric and Arithmetico-Geometric Progressions

Sum of an AP. The sum $S_n$ of the first $n$ terms of the AP $(a+(n-1)d)$ is

Sum of a GP. The sum $S_n$ of the first $n$ terms of the GP $(ar^{n-1})$ is

if $r=1$, the GP is simply the constant sequence $a,a,a,\ldots$, whose sum of the first $n$ terms is trivially $na$. Equivalently, for $r\ne1$: $1+r+r^2+\cdots+r^{n-1}=\dfrac{1-r^n}{1-r}$.

Sum of an AGP. A series is arithmetico-geometric if its terms form an AGP. The sum $S_n$ of the first $n$ terms of $((a+(n-1)d)r^{n-1})$, for $r\ne1$, is

$S_n = \frac{a-(a+(n-1)d)r^n}{1-r} + dr\left(\frac{1-r^{n-1}}{(1-r)^2}\right).$ …