In mathematics, an arithmetico-geometric sequence is the result of element-by-element multiplication of the elements of a geometric progression with the corresponding elements of an arithmetic progression. The nth element of an arithmetico-geometric sequence is the product of the nth element of an arithmetic sequence and the nth element of a geometric sequence. An arithmetico-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 of expected values in probability theory, especially in Bernoulli processes.

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 elements of this sequence has been called Gabriel's staircase and it has a value of 2. 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 ${\displaystyle u_{n+1}=ru_{n}+d}$, which combine the defining recurrence relations ${\displaystyle u_{n+1}=u_{n}+d}$ for arithmetic sequences and ${\displaystyle u_{n+1}=ru_{n}}$ for geometric sequences. These sequences are therefore solutions to a special class of linear difference equation: inhomogeneous first order linear recurrences with constant coefficients.

The elements of an arithmetico-geometric sequence ${\displaystyle (A_{n}G_{n})_{n\geq 1}}$ are the products of the elements of an arithmetic progression ${\displaystyle (A_{n})_{n\geq 1}}$ (in blue) with initial value ${\displaystyle a}$ and common difference ${\displaystyle d}$, ${\displaystyle A_{n}=a+(n-1)d,}$ with the corresponding elements of a geometric progression ${\displaystyle (G_{n})_{n\geq 1}}$ (in green) with initial value ${\displaystyle b}$ and common ratio ${\displaystyle r}$, ${\displaystyle G_{n}=br^{n-1},}$ so that

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

The sequence

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