Arithmetico Geometric Progression

Definition:

An arithmetico-geometric 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: ${a}_{1},\left({a}_{1}+{d}_{1}\right)r,\left({a}_{1}+2{d}_{1}\right){r}^{2},\dots$

• ${a}_{1}$ 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:

${a}_{n}={a}_{1}+\left(n-1\right){d}_{1}×{r}^{\left(n-1\right)}$

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:

${S}_{n}=\frac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}+\frac{{d}_{1}\left({r}^{n}-1\right)}{\left(r-1{\right)}^{2}}$

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 Arithmetico-Geometric Progression:

1. Combination of AP and GP:

• An AGP combines the concepts of both arithmetic and geometric progressions.
2. Complexity in Calculation:

• AGPs involve a mix of arithmetic and geometric sequences, making calculations more intricate compared to standard progressions.
3. 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.