# Geometric Progression

#### Definition:

A geometric progression (GP) is a sequence of numbers where each term after the first is obtained by multiplying the preceding term by a fixed, non-zero number called the "common ratio."

#### General Form:

A GP can be represented as: $a,ar,a{r}^{2},a{r}^{3},\dots$

• $a$ is the first term of the sequence.
• $r$ is the common ratio between consecutive terms.

#### nth Term of a GP:

The nth term of a GP is given by the formula:

${a}_{n}=a×{r}^{\left(n-1\right)}$

Where:

• ${a}_{n}$ is the nth term of the GP.
• $a$ is the first term of the GP.
• $r$ is the common ratio.
• $n$ is the term number.

#### Sum of First $n$ Terms of a GP:

The sum of the first $n$ terms of a GP (${S}_{n}$) can be calculated using the formula:

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

Where:

• ${S}_{n}$ is the sum of the first $n$ terms.
• $n$ is the number of terms.
• $a$ is the first term.
• $r$ is the common ratio.

#### Properties of Geometric Progression:

1. Common Ratio:

• In a GP, the ratio between consecutive terms is constant.
2. Nth Term:

• The formula to find the nth term of a GP is ${a}_{n}=a×{r}^{\left(n-1\right)}$.
3. Sum of a GP:

• The sum of the first $n$ terms (${S}_{n}$) can be calculated using ${S}_{n}=\frac{a×\left(1-{r}^{n}\right)}{1-r}$.
4. Relationship between Terms:

• Any term in a GP can be found by multiplying the common ratio to the previous term.
5. Infinite GP:

• If $\mathrm{\mid }r\mathrm{\mid }<1$, an infinite GP converges to $\frac{a}{1-r}$.
• If $\mathrm{\mid }r\mathrm{\mid }\ge 1$, the series diverges.

#### Applications of Geometric Progression:

• Finance: Compound interest calculations, growth and depreciation models, etc.
• Physics: Modelling exponential growth or decay, waveforms, etc.
• Computer Science: Algorithms involving geometric sequences, data compression, etc.