Arithmetico-geometric sequence

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In mathematics, an arithmetico-geometric sequence is the result of the multiplication of a geometric progression with the corresponding terms of an arithmetic progression.

Sequence, nth term

The sequence has the terms;

$[a+(n-1)d] r^{n-1}$

where r is the common ratio, and the coefficients of r^{n − 1};

$[a+(n-1)d]$

are terms from the arithmetic progression with difference d and initial value a.

Series, sum to n terms

An arithmetico-geometric series has the form

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

and the sum to n terms is equal to;

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

Derivation

Starting from the series,

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

multiply S_n by r,

$r S_n = ar+(a+d)r^2+(a+2d)r^3+\cdots +(a+(n-1)d)r^n$

subtract rS_n from S_n,

$\begin{align} S_n(1-r) &=&\left[a+(a+d)r+(a+2d)r^2+\cdots +[a+d(n-1)]r^{n-1}\right] \\ & &- \left[ar+(a+d)r^2+(a+2d)r^3+\cdots + [ a+d(n-1) ] r^n\right] \\ & = & a+ \left[ rd + r^2d + \cdots \right] - [ a+d(n-1) ] r^n \\ & = & a + \left[ \frac{rd(1-r^{n-1})}{1-r}\right]-[a+(n-1)d]r^n\end{align}$

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

$\lim_{n \to \infty}S_{n} = \frac{a}{1-r}+\frac{dr}{(1-r)^2}$

If r is outside of the above range, the series either diverges or alternates.

See also

• partial sum