Arithmetico-geometric sequence

Not to be confused with Arithmetic–geometric mean.

In mathematics, an arithmetico-geometric sequence is the result of the multiplication of a geometric progression with the corresponding terms of an arithmetic progression. It should be noted that the corresponding French term refers to a different concept (sequences of the form ) which is a special case of linear difference equations.

Sequence, nth term

The sequence has the nth term defined for n ≥ 1 as: are terms from the arithmetic progression with difference d and initial value a and geometric progression with initial value "b" and common ratio "r"

Series, sum to n terms

An arithmetico-geometric series has the form and the sum to n terms is equal to: Derivation

Starting from the series, multiply Sn by r, subtract rSn from Sn, using the expression for the sum of a geometric series in the middle series of terms. Finally dividing through by (1 − r) gives the result.

Sum to infinite terms

If −1 < r < 1, then the sum of the infinite number of terms of the progression is If r is outside of the above range, the series either

• diverges (when r > 1, or when r = 1 where the series is arithmetic and a and d are not both zero; if both a and d are zero in the later case, all terms of the series are zero and the series is constant)
• or alternates (when r ≤ −1).