Suppose \({a_1},{a_2},{a_3}, \ldots  \ldots  \ldots {a_n}\) is an \(A.P.\), and \({b_1},{b_2},{b_3},{\rm{ }} \ldots  \ldots ..{b_n}\) is a \(G.P.\), then the sequence \({a_1}{b_1},{a_2}{b_2},{a_3}{b_3},{\rm{ }} \ldots  \ldots .{a_n}{b_n}\) is said to be an arithmetic – geometric sequence.  

Here are certain examples of Arithmetic-geometric series:

\(S = {\rm{ }}1{\rm{ }} + {\rm{ }}2x + {\rm{ }}3{x^2} + {\rm{ }}4{x^3} + {\rm{ }} \ldots  \ldots  \ldots {\rm{ }} + {\rm{ }}\infty \) terms

where \(x,{x^2},{x^3},{\rm{ }} \ldots .\) Are in \(GP\) and \(1, 2, 3, ….\) are in AP.