# Definition:Geometric Sequence

## Definition

A **geometric sequence** is a sequence $\sequence {x_n}$ 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:

- $x_n = \begin{cases} a & : n = 0 \\ r x_{n-1} & : n > 0 \\ \end{cases}$

### Term

The elements:

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

are the **terms** of $\sequence {x_n}$.

### 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 $\sequence {x_n}$.

## Finite Geometric Sequence

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

### Initial Term

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

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

The same definition applies to a finite geometric sequence $G_n = \sequence {a_0, a_1, \ldots, a_n}$.

## Also known as

The usual term is **geometric progression**, and the abbreviation **G.P.** is often seen.

However, $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers the term **sequence** as there is less likelihood of confusing it with **geometric series**, which the term **geometric progression** is also often used for.

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 sequences**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 - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**geometric progression** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**geometric progression (geometric sequence)** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**geometric progression (geometric sequence)** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**geometric sequence**