Definition:Geometric Sequence

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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:

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


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.