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, nonzero 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:
${a}_{n}=a\times {r}^{(n1)}$
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:
${S}_{n}=\frac{a\times (1{r}^{n})}{1r}$
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:

Common Ratio:
 In a GP, the ratio between consecutive terms is constant.

Nth Term:
 The formula to find the nth term of a GP is ${a}_{n}=a\times {r}^{(n1)}$.

Sum of a GP:
 The sum of the first $n$ terms (${S}_{n}$) can be calculated using ${S}_{n}=\frac{a\times (1{r}^{n})}{1r}$.

Relationship between Terms:
 Any term in a GP can be found by multiplying the common ratio to the previous term.

Infinite GP:
 If $\mathrm{\mid}r\mathrm{\mid}<1$, an infinite GP converges to $\frac{a}{1r}$.
 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.