Q.Find $S_n$ of the following arithmetico-geometric sequence: $2, 4x, 6x^2, 8x^3, 10x^4, \ldots$

An Arithmetico-Geometric Progression (A.G.P.) is a sequence in which each term is the product of the corresponding terms of an A.P. and a G.P.: if the A.P. is $a, a+d, a+2d, \ldots$ and the G.P. is $1, r, r^2, \ldots$, the A.G.P. is $a, (a+d)r, (a+2d)r^2, \ldots$, so its general term is $t_n=[a+(n-1)d]\,r^{n-1}$. The sum of the first $n$ terms is found by the same shift-and-subtract trick used for a plain G.P.: write $S_n$, then write $rS_n$ (every term shifted one place), and subtract; because the A.P. part now differs by a constant $d$, the leftover terms form a genuine G.P. that can be summed with the usual formula, giving $S_n=\dfrac{a}{1-r}+\dfrac{dr(1-r^{n-1})}{(1-r)^2}-\dfrac{[a+(n-1)d]r^n}{1-r}$ for $r\ne1$. When $|r|<1$, letting $n\to\infty$ makes the last term vanish, giving the simpler sum to infinity $\dfrac{a}{1-r}+\dfrac{dr}{(1-r)^2}$.