In mathematics, an arithmetico-geometric sequence is the result of the term-by-term 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
is an arithmetico-geometric sequence. The arithmetic component appears in the numerator (in blue), and the geometric one in the denominator (in green).
The summation of this infinite sequence is known as a arithmetico-geometric series, and its most basic form has been called Gabriel's staircase:
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:
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 closed-form expression
by r, gives
Subtracting rSn from Sn, 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.
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
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
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