Arithmetico-geometric Sequence
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Arithmetico-geometric Sequence

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:[1][2][3]

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:[4]

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 closed-form expression

Proof

Multiplying,[4]

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.

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[4]

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

.

References

  1. ^ Swain, Stuart G. (2018). "Proof Without Words: Gabriel's Staircase". Mathematics Magazine. 67 (3): 209-209. doi:10.1080/0025570X.1994.11996214. ISSN 0025-570X.
  2. ^ Weisstein, Eric W. "Gabriel's Staircase". MathWorld.
  3. ^ Edgar, Tom (2018). "Staircase Series". Mathematics Magazine. 91 (2): 92-95. doi:10.1080/0025570X.2017.1415584. ISSN 0025-570X.
  4. ^ 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


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