Arithmetico-Geometric Progression (AGP)

Combining an arithmetic progression and a geometric progression term-by-term produces a new kind of progression.

Definition 5.1. A sequence of the form

is called an arithmetico-geometric progression (AGP). Concretely: take an AP $a,a+d,a+2d,\ldots$ and a GP $1,r,r^2,\ldots$, and multiply them term-by-term.

AGPs arise naturally in applications such as computing an expected value in probability theory. The $n^{th}$ term is

Setting $r=1$ collapses the AGP to a plain AP; setting $d=0$ collapses it to a plain GP — so every AP and every GP is itself a (degenerate) AGP, and the AGP framework is the genuine generalisation. …