Arithmetico-geometric sequence should not be confused with Arithmetic–geometric mean.
In mathematics, arithmetico-geometric sequence is the result of term-by-term multiplication of a geometric progression with the corresponding terms of an arithmetic progression. Put plainly, the nth term of an arithmetico-geometric sequence is the product of the nth term of an arithmetic sequenceand the nth term of a geometric one.[1] Arithmetico-geometric sequences arise in various applications, such as the computation of expected values in probability theory. For instance, the sequence
\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}}, …
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 an arithmetico-geometric series, and its most basic form has been called Gabriel's staircase:[2] [3]
infty | |
\sum | |
k=1 |
{\color{blue}k}{\color{green}rk}=
r | |
(1-r)2 |
, for 0<r<1
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
un+1=aun+b
The first few terms of an arithmetico-geometric sequence composed of an arithmetic progression (in blue) with difference
d
a
b
r
\begin{align} t1&=\color{blue}a\color{green}b\\ t2&=\color{blue}(a+d)\color{green}br\\ t3&=\color{blue}(a+2d)\color{green}br2\\ & \vdots\\ tn&=\color{blue}[a+(n-1)d]\color{green}brn-1\end{align}
For instance, the sequence
\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}}, …
is defined by
d=b=1
a=0
r=1/2
The sum of the first terms of an arithmetico-geometric sequence has the form
\begin{align} Sn&=
n | |
\sum | |
k=1 |
tk=
n | |
\sum | |
k=1 |
\left[a+(k-1)d\right]brk\\ &=ab+[a+d]br+[a+2d]br2+ … +[a+(n-1)d]brn\\ &=A1G1+A2G2+A3G3+ … +AnGn, \end{align}
Ai
Gi
This sum has the closed-form expression
\begin{align} Sn&=
ab-(a+nd)brn | + | |
1-r |
dbr(1-rn) | |
(1-r)2 |
\\ &=
A1G1-An+1Gn+1 | + | |
1-r |
dr | |
(1-r)2 |
(G1-Gn+1). \end{align}
Multiplying,[4]
Sn=a+[a+d]r+[a+2d]r2+ … +[a+(n-1)d]rn
rSn=ar+[a+d]r2+[a+2d]r3+ … +[a+(n-1)d]rn.
Subtracting from, and using the technique of telescoping series gives
\begin{align}(1-r)Sn={}&\left[a+(a+d)r+(a+2d)r2+ … +[a+(n-1)d]rn\right]\\[5pt] &{}-\left[ar+(a+d)r2+(a+2d)r3+ … +[a+(n-1)d]rn\right]\\[5pt] ={}&a+d\left(r+r2+ … +rn-1\right)-\left[a+(n-1)d\right]rn\\[5pt] ={}&a+d\left(r+r2+ … +rn-1+rn\right)-\left(a+nd\right)rn\\[5pt] ={}&a+dr\left(1+r+r2+ … +rn-1\right)-\left(a+nd\right)rn\\[5pt] ={}&a+
dr(1-rn) | |
1-r |
-(a+nd)rn,\end{align}
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]
\begin{align} S&=
infty | |
\sum | |
k=1 |
tk=\limnSn\\ &=
ab | + | |
1-r |
dbr | |
(1-r)2 |
\\ &=
A1G1 | + | |
1-r |
dG1r | |
(1-r)2 |
. \end{align}
If r is outside of the above range, the series either
For instance, the sum
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}}+ …
being the sum of an arithmetico-geometric series defined by
d=b=1
a=0
r= | 12 |
S=2
This sequence corresponds to the expected number of coin tosses before obtaining "tails". The probability
Tk
T | ||||
|
|
k=
1{2 | |
k} |
Therefore, the expected number of tosses is given by
infty | |
\sum | |
k=1 |
kTk=
infty | |
\sum | |
k=1 |
\color{blue | |
k}{\color{green}2 |
k}=S=2