In mathematics, an arithmetico-geometric sequence is the result of term-by-term multiplication of a geometric progression with the corresponding terms of an arithmetic progression. 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 sequence.[1] 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 series summation of the infinite terms of this sequence is known as an arithmetico-geometric series, and it has been called Gabriel's staircase.[2][3] In general,
The label of arithmetico-geometric sequence may also be given to different objects combining characteristics of both arithmetic and geometric sequences. For instance, the French notion of arithmetico-geometric sequence refers to sequences that satisfy recurrence relations of the form , which combine the recurrence relations for arithmetic sequences and for geometric sequences. These sequences are therefore solutions to a special class of linear difference equation: inhomogeneous first order linear recurrences with constant coefficients.
Terms of the sequence
editThe 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
editFor instance, the sequence
is defined by , , and .
Sum of the terms
editThe 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
editMultiplying,[4]
by r, gives
Subtracting rSn from Sn, and using the technique of telescoping series gives
where the last equality follows from the formula for the sum of a finite geometric series. Finally, rearranging to group the first and third terms of that expression, rewriting each group in terms of and and then dividing through by 1 − r gives the closed-form expression claimed above.
Infinite series
editIf −1 < r < 1, then the sum S of the arithmetico-geometric series, that is to say, the limit of the partial sums of all the infinitely many terms of the progression, is given by[4]
If r is outside of the above range, b is not zero, and a and d are not both zero, the series diverges.
Example: application to expected values
editThe sum
- ,
is the sum of an arithmetico-geometric series defined by , , and , and it converges to . This sequence corresponds to the expected number of coin tosses required to obtain "tails". The probability of obtaining tails for the first time at the kth toss is as follows:
- .
Therefore, the expected number of tosses to reach the first "tails" is given by
- .
References
edit- ^ "Arithmetic-Geometric Progression | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2021-04-21.
- ^ Swain, Stuart G. (2018). "Proof Without Words: Gabriel's Staircase". Mathematics Magazine. 67 (3): 209. doi:10.1080/0025570X.1994.11996214. ISSN 0025-570X.
- ^ Edgar, Tom (2018). "Staircase Series". Mathematics Magazine. 91 (2): 92–95. doi:10.1080/0025570X.2017.1415584. ISSN 0025-570X. S2CID 218542483.
- ^ 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
edit- D. Khattar. The Pearson Guide to Mathematics for the IIT-JEE, 2/e (New ed.). 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.