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

${\displaystyle {\dfrac {\color {blue}{0}}{\color {green}{1}}},\ {\dfrac {\color {blue}{1}}{\color {green}{2}}},\ {\dfrac {\color {blue}{2}}{\color {green}{4}}},\ {\dfrac {\color {blue}{3}}{\color {green}{8}}},\ {\dfrac {\color {blue}{4}}{\color {green}{16}}},\ {\dfrac {\color {blue}{5}}{\color {green}{32}}},\cdots }$

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]

${\displaystyle \sum _{k=1}^{\infty }{\color {blue}k}{\color {green}r^{k}}={\frac {r}{(1-r)^{2}}},\quad \mathrm {for\ } 0

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 ${\displaystyle u_{n+1}=au_{n}+b}$, 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 ${\displaystyle d}$ and initial value ${\displaystyle a}$ and a geometric progression (in green) with initial value ${\displaystyle b}$ and common ratio ${\displaystyle r}$ are given by:[4]

{\displaystyle {\begin{aligned}t_{1}&=\color {blue}a\color {green}b\\t_{2}&=\color {blue}(a+d)\color {green}br\\t_{3}&=\color {blue}(a+2d)\color {green}br^{2}\\&\ \,\vdots \\t_{n}&=\color {blue}[a+(n-1)d]\color {green}br^{n-1}\end{aligned}}}

### Example

For instance, the sequence

${\displaystyle {\dfrac {\color {blue}{0}}{\color {green}{1}}},\ {\dfrac {\color {blue}{1}}{\color {green}{2}}},\ {\dfrac {\color {blue}{2}}{\color {green}{4}}},\ {\dfrac {\color {blue}{3}}{\color {green}{8}}},\ {\dfrac {\color {blue}{4}}{\color {green}{16}}},\ {\dfrac {\color {blue}{5}}{\color {green}{32}}},\cdots }$

is defined by ${\displaystyle d=b=1}$, ${\displaystyle a=0}$, and ${\displaystyle r={\frac {1}{2}}}$.

## Sum of the terms

The sum of the first n terms of an arithmetico-geometric sequence has the form

{\displaystyle {\begin{aligned}S_{n}&=\sum _{k=1}^{n}t_{k}=\sum _{k=1}^{n}\left[a+(k-1)d\right]br^{k-1}\\&=ab+[a+d]br+[a+2d]br^{2}+\cdots +[a+(n-1)d]br^{n-1}\\&=A_{1}G_{1}+A_{2}G_{2}+A_{3}G_{3}+\cdots +A_{n}G_{n},\end{aligned}}}

where ${\displaystyle A_{i}}$ and ${\displaystyle G_{i}}$ are the ith terms of the arithmetic and the geometric sequence, respectively.

This sum has the closed-form expression

{\displaystyle {\begin{aligned}S_{n}&={\frac {ab-(a+nd)\,br^{n}}{1-r}}+{\frac {dbr\,(1-r^{n})}{(1-r)^{2}}}\\&={\frac {A_{1}G_{1}-A_{n+1}G_{n+1}}{1-r}}+{\frac {dr}{(1-r)^{2}}}\,(G_{1}-G_{n+1}).\end{aligned}}}

### Proof

Multiplying,[4]

${\displaystyle S_{n}=ab+[a+d]br+[a+2d]br^{2}+\cdots +[a+(n-1)d]br^{n-1}}$

by r, gives

${\displaystyle rS_{n}=abr+[a+d]br^{2}+[a+2d]br^{3}+\cdots +[a+(n-1)d]br^{n}.}$

Subtracting rSn from Sn, and using the technique of telescoping series gives

{\displaystyle {\begin{aligned}(1-r)S_{n}={}&\left[ab+(a+d)br+(a+2d)br^{2}+\cdots +[a+(n-1)d]br^{n-1}\right]\\[5pt]&{}-\left[abr+(a+d)br^{2}+(a+2d)br^{3}+\cdots +[a+(n-1)d]br^{n}\right]\\[5pt]={}&ab+db\left(r+r^{2}+\cdots +r^{n-1}\right)-\left[a+(n-1)d\right]br^{n}\\[5pt]={}&ab+db\left(r+r^{2}+\cdots +r^{n-1}+r^{n}\right)-\left(a+nd\right)br^{n}\\[5pt]={}&ab+dbr\left(1+r+r^{2}+\cdots +r^{n-1}\right)-\left(a+nd\right)br^{n}\\[5pt]={}&ab+{\frac {dbr(1-r^{n})}{1-r}}-(a+nd)br^{n},\end{aligned}}}

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]

{\displaystyle {\begin{aligned}S&=\sum _{k=1}^{\infty }t_{k}=\lim _{n\to \infty }S_{n}\\&={\frac {ab}{1-r}}+{\frac {dbr}{(1-r)^{2}}}\\&={\frac {A_{1}G_{1}}{1-r}}+{\frac {dG_{1}r}{(1-r)^{2}}}.\end{aligned}}}

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

${\displaystyle S={\dfrac {\color {blue}{0}}{\color {green}{1}}}+{\dfrac {\color {blue}{1}}{\color {green}{2}}}+{\dfrac {\color {blue}{2}}{\color {green}{4}}}+{\dfrac {\color {blue}{3}}{\color {green}{8}}}+{\dfrac {\color {blue}{4}}{\color {green}{16}}}+{\dfrac {\color {blue}{5}}{\color {green}{32}}}+\cdots }$,

being the sum of an arithmetico-geometric series defined by ${\displaystyle d=b=1}$, ${\displaystyle a=0}$, and ${\displaystyle r={\frac {1}{2}}}$, converges to ${\displaystyle S=2}$.

This sequence corresponds to the expected number of coin tosses before obtaining "tails". The probability ${\displaystyle T_{k}}$ of obtaining tails for the first time at the kth toss is as follows:

${\displaystyle T_{1}={\frac {1}{2}},\ T_{2}={\frac {1}{4}},\dots ,T_{k}={\frac {1}{2^{k}}}}$.

Therefore, the expected number of tosses is given by

${\displaystyle \sum _{k=1}^{\infty }kT_{k}=\sum _{k=1}^{\infty }{\frac {\color {blue}k}{\color {green}2^{k}}}=S=2}$ .

## 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. ^
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.