|Part of a series of articles about|
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
Series, sum to n terms
An arithmetico-geometric series has the form
and the sum to n terms is equal to:
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 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).
- 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.
- D. Khattar. The Pearson Guide to Mathematics for the IIT-JEE, 2/e (New Edition). Pearson Education India. p. 10.8. ISBN 81-317-2876-5.
- P. Gupta. Comprehensive Mathematics XI. Laxmi Publications. p. 380. ISBN 81-7008-597-7.