Arithmetico Geometric Progression
Definition:
An arithmeticogeometric progression (AGP) is a sequence where each term is determined by a combination of both an arithmetic progression (AP) and a geometric progression (GP).
General Form:
An AGP is represented as a sequence where each term is obtained by adding an arithmetic progression to a geometric progression.
The terms of an AGP can be expressed as: ${a}_{1},({a}_{1}+{d}_{1})r,({a}_{1}+2{d}_{1}){r}^{2},\dots $
 ${a}_{1}$ is the first term of the AP.
 ${d}_{1}$ is the common difference of the AP.
 $r$is the common ratio of the GP.
nth Term of an AGP:
The nth term of an AGP is calculated by the formula:
${a}_{n}={a}_{1}+(n1){d}_{1}\times {r}^{(n1)}$
Where:
 ${a}_{n}$ is the nth term of the AGP.
 ${a}_{1}$ is the first term of the AP.
 ${d}_{1}$ is the common difference of the AP.
 $r$ is the common ratio of the GP.
 $n$ is the term number.
Sum of First $n$ Terms of an AGP:
The sum of the first $n$ terms of an AGP (${S}_{n}$) can be calculated using the formula:
${S}_{n}=\frac{{a}_{1}(1{r}^{n})}{1r}+\frac{{d}_{1}({r}^{n}1)}{(r1{)}^{2}}$
Where:
 ${S}_{n}$ is the sum of the first $n$ terms.
 $n$ is the number of terms.
 ${a}_{1}$ is the first term of the AP.
 ${d}_{1}$ is the common difference of the AP.
 $r$ is the common ratio of the GP.
Properties of ArithmeticoGeometric Progression:

Combination of AP and GP:
 An AGP combines the concepts of both arithmetic and geometric progressions.

Complexity in Calculation:
 AGPs involve a mix of arithmetic and geometric sequences, making calculations more intricate compared to standard progressions.

Applications:
 Used in various mathematical problems where both arithmetic and geometric relationships coexist.
Applications of AGP:
 Mathematics:
 In calculus for finding definite integrals involving AGPs.
 In number theory for special sequences.
 Physics:
 Models oscillatory systems, waveforms, etc.