Sequence, nth term
The sequence has the nth term defined for n ≥ 1 as:
where r is the common ratio, and the coefficients of rn − 1
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
and the sum to n terms is equal to:
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).