Arithmetico–geometric sequence
In mathematics, an arithmetico–geometric sequence is the result of the termbyterm multiplication of a geometric progression with the corresponding terms of an arithmetic progression. Put more plainly, the nth term of an arithmetico–geometric sequence is the product of the nth term of an arithmetic sequence and the nth term of a geometric one. Arithmetico–geometric sequences arise in various applications, such as the computation of expected values in probability theory. For instance, the sequence
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Specialized 

is an arithmetico–geometric sequence. The arithmetic component appears in the numerator (in blue), and the geometric one in the denominator (in green).
The denomination may also be applied to different objects presenting characteristics of both arithmetic and geometric sequences; for instance the French notion of arithmetico–geometric sequence refers to sequences of the form , which generalise both arithmetic and geometric sequences. Such sequences are a special case of linear difference equations.
Terms of the sequence
The first few terms of an arithmetico–geometric sequence composed of an arithmetic progression (in blue) with difference and initial value and a geometric progression (in green) with initial value and common ratio are given by:[1]
Example
For instance, the sequence
is defined by , , and .
Sum of the terms
The sum of the first n terms of an arithmetico–geometric sequence has the form
where and are the ith terms of the arithmetic and the geometric sequence, respectively.
This sum has the closedform expression
Proof
Multiplying,[1]
by r, gives
Subtracting rS_{n} from S_{n}, and using the technique of telescoping series gives
where the last equality results of the expression for the sum of a geometric series. Finally dividing through by 1 − r gives the result.
Infinite series
If −1 < r < 1, then the sum S of the arithmetico–geometric series, that is to say, the sum of all the infinitely many terms of the progression, is given by[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).
Example: application to expected values
For instance, the sum
 ,
being the sum of an arithmetico–geometric series defined by , , and , converges to .
This sequence corresponds to the expected number of coin tosses before obtaining "tails". The probability of obtaining tails for the first time at the kth toss is as follows:
 .
Therefore, the expected number of tosses is given by
 .
References
 K. F. Riley; M. P. Hobson; S. J. Bence (2010). Mathematical methods for physics and engineering (3rd ed.). Cambridge University Press. p. 118. ISBN 9780521861533.
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.