Arithmetico-geometric sequence

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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[edit]

The sequence has the nth term[1] 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[edit]

An arithmetico-geometric series has the form

and the sum to n terms is equal to:


Starting from the series,[1]

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[edit]

If −1 < r < 1, then the sum of the infinite number of terms of the progression is[1]

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).

See also[edit]


  1. ^ a b c K.F. Riley; M.P. Hobson; S.J. Bence (2010). Mathematical methods for physics and engineering (3rd ed.). Cambridge University Press. p. 118. ISBN 978-0-521-86153-3. 

Further reading[edit]