Arithmetico-Geometric Progression (A.G.P.)

A sequence in which each term is the product of the corresponding terms of an A.P. and a G.P. is called an Arithmetico-Geometric Progression (A.G.P.). E.g. consider the A.P. $2,5,8,11$ (first term $a=2$, common difference $d=3$) written as $(a),(a+d),(a+2d),(a+3d)$ and the G.P. $1,3,9,27$ (first term 1, common ratio $r=3$) written as $(1),(r),(r^2),(r^3)$; multiplying term-by-term gives the A.G.P. $a,(a+d)r,(a+2d)r^2,(a+3d)r^3,\ldots$, i.e. $2\times1,5\times3,8\times9,11\times27,\ldots$ Here the first factor of each term follows the A.P. and the second factor follows the G.P., so the sequence forms an A.G.P. …