Arithmetico-geometric sequence

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 nth term[1] defined for n ≥ 1 as:

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

where r is the common ratio, and the coefficients of rn − 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

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

and the sum to n terms is equal to:

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

Derivation

Starting from the series,[1]

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

multiply Sn by r,

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

subtract rSn from Sn,

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

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

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).