# 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 ${\displaystyle u_{n+1}=au_{n}+b}$) which is a special case of linear difference equations.

## Sequence, nth term

The sequence has the nth term[1] defined for n ≥ 1 as:

${\displaystyle [a+(n-1)d]br^{n-1}}$

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

${\displaystyle \sum _{k=1}^{n}\left[a+(k-1)d\right]r^{k-1}=a+[a+d]r+[a+2d]r^{2}+\cdots +[a+(n-1)d]r^{n-1}}$

and the sum to n terms is equal to:

${\displaystyle S_{n}=\sum _{k=1}^{n}\left[a+(k-1)d\right]r^{k-1}={\frac {a-[a+(n-1)d]r^{n}}{1-r}}+{\frac {dr(1-r^{n-1})}{(1-r)^{2}}}.}$

### Derivation

Starting from the series,[1]

${\displaystyle S_{n}=a+[a+d]r+[a+2d]r^{2}+\cdots +[a+(n-1)d]r^{n-1}}$

multiply Sn by r,

${\displaystyle rS_{n}=ar+[a+d]r^{2}+[a+2d]r^{3}+\cdots +[a+(n-1)d]r^{n}}$

subtract rSn from Sn,

{\displaystyle {\begin{aligned}(1-r)S_{n}&=&\left[a+(a+d)r+(a+2d)r^{2}+\cdots +[a+(n-1)d]r^{n-1}\right]\\&&-\left[ar+(a+d)r^{2}+(a+2d)r^{3}+\cdots +[a+(n-1)d]r^{n}\right]\\&=&a+d\left(r+r^{2}+\cdots +r^{n-1}\right)-\left[a+(n-1)d\right]r^{n}\\&=&a+{\frac {dr(1-r^{n-1})}{1-r}}-[a+(n-1)d]r^{n}\end{aligned}}}

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[1]

${\displaystyle \lim _{n\to \infty }S_{n}={\frac {a}{1-r}}+{\frac {rd}{(1-r)^{2}}}}$

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).