Definition:Geometric Progression

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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:

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


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.