# Definition:Geometric Progression

## Contents

## Definition

A **geometric progression** is a sequence $\left \langle{x_n}\right \rangle$ in $\R$ defined as:

- $x_n = a r^n$ for $n = 0, 1, 2, 3, \ldots$

Thus its general form is:

- $a, ar, ar^2, ar^3, \ldots$

and the general term can be defined recursively as:

- $x_n = \begin{cases} a & : n = 0 \\ r x_n & : n > 0 \\ \end{cases}$

### Term

The elements:

- $x_n$ for $n = 0, 1, 2, 3, \ldots$

are the **terms** of $\left \langle{x_n}\right \rangle$.

### Common Ratio

The parameter:

- $r \in \R: r \ne 0$

is called the **common ratio** of $\sequence {x_n}$.

### Scale Factor

The parameter:

- $a \in \R: a \ne 0$

is called the **scale factor** of $\left \langle{x_n}\right \rangle$.

## Finite Geometric Progression

A **finite geometric progression** is a geometric progression with a finite number of terms .

### Initial Term

Let $G = \left\langle{a_0, a_1, \ldots}\right\rangle$ be a geometric progression.

The **initial term** of $G_n$ is the term $a_0$.

The same definition applies to a finite geometric progression $G_n = \left\langle{a_0, a_1, \ldots, a_n}\right\rangle$.

## Also known as

Euclid used the term **continued proportion** throughout Book $\text{VIII}$ of *The Elements*, though never formally defining it.

In the words of Euclid:

*If there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, the numbers are the least of those which have the same ratio with them.*

(*The Elements*: Book $\text{VIII}$: Proposition $1$)

## Also see

- Results about
**geometric progressions**can be found here.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.2$: More about Numbers: Irrationals, Perfect Numbers and Mersenne Primes - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**geometric progression (geometric sequence)**