Arithmeticogeometric sequence
Part of a series of articles about  
Calculus  



Specialized 

In mathematics, an arithmeticogeometric 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
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
An arithmeticogeometric series has the form
and the sum to n terms is equal to:
Derivation
Starting from the series,^{[1]}
multiply S_{n} by r,
subtract rS_{n} from S_{n},
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]}
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
References
Further reading
 D. Khattar. The Pearson Guide to Mathematics for the IITJEE, 2/e (New Edition). Pearson Education India. p. 10.8. ISBN 8131728765.
 P. Gupta. Comprehensive Mathematics XI. Laxmi Publications. p. 380. ISBN 8170085977.