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Inmathematics, anarithmetico-geometric sequence is the result of element-by-element multiplication of the elements of ageometric progression with the corresponding elements of anarithmetic progression. Thenth element of an arithmetico-geometric sequence is the product of thenth element of an arithmetic sequence and thenth element of a geometric sequence.^[1] Anarithmetico-geometric series is a sum of terms that are the elements of an arithmetico-geometric sequence. Arithmetico-geometric sequences and series arise in various applications, such as the computation ofexpected values inprobability theory, especially inBernoulli processes.

For instance, the sequence

${\displaystyle \frac{0}{1},\frac{1}{2},\frac{2}{4},\frac{3}{8},\frac{4}{16},\frac{5}{32},\cdots }$

is an arithmetico-geometric sequence. The arithmetic component appears in the numerator (inblue), and the geometric one in the denominator (ingreen). Theseries summation of the infinite elements of this sequence has been calledGabriel's staircase and it has a value of 2.^[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, theFrench notion of arithmetico-geometric sequence refers to sequences that satisfyrecurrence relations of the form${\displaystyle u_{n+1}=r{u}_n+d}$, which combine the defining recurrence relations${\displaystyle u_{n+1}=u_n+d}$ for arithmetic sequences and${\displaystyle u_{n+1}=r{u}_n}$ for geometric sequences. These sequences are therefore solutions to a special class oflinear difference equation: inhomogeneous first orderlinear recurrences with constant coefficients.

The elements of an arithmetico-geometric sequence${\displaystyle (A_n{G}_n{)}_{n\ge 1}}$ are the products of the elements of anarithmetic progression${\displaystyle (A_n{)}_{n\ge 1}}$ (in blue) with initial value${\displaystyle a}$ and common difference${\displaystyle d}$,${\displaystyle A_n=a+(n-1)d,}$ with the corresponding elements of ageometric progression${\displaystyle (G_n{)}_{n\ge 1}}$ (in green) with initial value${\displaystyle b}$ and common ratio${\displaystyle r}$,${\displaystyle G_n=b{r}^{n-1},}$ so that^[4]

These four parameters are somewhat redundant and can be reduced to three:${\displaystyle ab,}$${\displaystyle bd,}$ and${\displaystyle r.}$

The sequence

${\displaystyle \frac{0}{1},\frac{1}{2},\frac{2}{4},\frac{3}{8},\frac{4}{16},\frac{5}{32},\cdots }$

is the arithmetico-geometric sequence with parameters${\displaystyle d=b=1}$,${\displaystyle a=0}$, and${\displaystyle r=\frac{1}{2}}$.

The sum of the firstn terms of an arithmetico-geometric series has the form

where$A_i$ and$G_i$ are theith elements of the arithmetic and the geometric sequence, respectively.

This partial sum has theclosed-form expression

Multiplying^[4]

${\displaystyle S_n=ab+(a+d)br+(a+2d)b{r}^2+\cdots +(a+(n-1)d)b{r}^{n-1}}$

byr gives

${\displaystyle r{S}_n=abr+(a+d)b{r}^2+(a+2d)b{r}^3+\cdots +(a+(n-1)d)b{r}^n.}$

SubtractingrS_n fromS_n, dividing both sides by${\displaystyle b}$, and using the technique oftelescoping series (second equality) and the formula for the sum of a finitegeometric series (fifth equality) gives

as claimed.

If−1 <r < 1, then the sumS of the arithmetico-geometricseries, that is to say, thelimit of the partial sums of the elements of the sequence, is given by^[4]

Ifr is outside of the above range,b is not zero, anda andd are not both zero, the limit does not exist and the series isdivergent.

The sum

${\displaystyle S=\frac{0}{1}+\frac{1}{2}+\frac{2}{4}+\frac{3}{8}+\frac{4}{16}+\frac{5}{32}+\cdots }$,

is the sum of an arithmetico-geometric series defined by${\displaystyle d=b=1}$,${\displaystyle a=0}$, and${\displaystyle r=\frac{1}{2}}$, and it converges to${\displaystyle S=2}$. This sequence corresponds to the expected number ofcoin tosses required to obtain "tails". The probability${\displaystyle T_k}$ of obtaining tails for the first time at thekth 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 to reach the first "tails" is given by

${\displaystyle \sum _{k=1}^{\mathrm{\infty }}k{T}_k=\sum _{k=1}^{\mathrm{\infty }}\frac{k}{2^k}=2.}$

Similarly, the sum

${\displaystyle S=\frac{0\cdot \frac{1}{6}}{\frac{5}{6}}+\frac{1\cdot \frac{1}{6}}{1}+\frac{2\cdot \frac{1}{6}}{\frac{6}{5}}+\frac{3\cdot \frac{1}{6}}{{\left(\frac{6}{5}\right)}^2}+\frac{4\cdot \frac{1}{6}}{{\left(\frac{6}{5}\right)}^3}+\frac{5\cdot \frac{1}{6}}{{\left(\frac{6}{5}\right)}^4}+\cdots }$

is the sum of an arithmetico-geometric series defined by${\displaystyle d=1}$,${\displaystyle a=0}$,${\displaystyle b=\frac{1/6}{5/6}=\frac{1}{5}}$, and${\displaystyle r=\frac{5}{6}}$, and it converges to 6. This sequence corresponds to the expected number ofsix-sided dice rolls required to obtain a specific value on a die roll, for instance "5". In general, these series with${\displaystyle d=1}$,${\displaystyle a=0}$,${\displaystyle b=\frac{p}{1-p}}$, and${\displaystyle r=1-p}$ give the expectations of "the number of trials until first success" inBernoulli processes with "success probability"${\displaystyle p}$. The probabilities of each outcome follow ageometric distribution and provide the geometric sequence factors in the terms of the series, while the number of trials per outcome provides the arithmetic sequence factors in the terms.

• 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.

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