Citizendium - building a quality free general knowledge encyclopedia. Click here to join and contribute—free Many thanks December donors; special to Darren Duncan. January donations open; need minimum total \$100. Let's exceed that. Donate here. By donating you gift yourself and CZ.

# Geometric sequence

Main Article
Talk
Related Articles  [?]
Bibliography  [?]
Citable Version  [?]

This editable Main Article has an approved citable version (see its Citable Version subpage). While we have done conscientious work, we cannot guarantee that this Main Article, or its citable version, is wholly free of mistakes. By helping to improve this editable Main Article, you will help the process of generating a new, improved citable version.

## Contents

A geometric sequence (or geometric progression) is a (finite or infinite) sequence of (real or complex) numbers such that the quotient (or ratio) of consecutive elements is the same for every pair.

In finance, compound interest generates a geometric sequence.

## Examples

Examples for geometric sequences are

• $3, 6, 12, 24, 48, 96$ (finite, length 6: 6 elements, quotient 2)
• $1, -2, 4, -8$ (finite, length 4: 4 elements, quotient −2)
• $8, 4, 2, 1, {1\over2}, {1\over4}, {1\over8}, \dots {1\over2^{n-4}}, \dots$ (infinite, quotient $1\over2$)
• $2, 2, 2, 2, \dots$ (infinite, quotient 1)
• $-2, 2, -2, 2, \dots , (-1)^n\cdot 2 , \dots$ (infinite, quotient −1)
• ${1\over2}, 1, 2, 4, \dots , 2^{n-2}, \dots$ (infinite, quotient 2)
• $1, 0, 0, 0, \dots \$ (infinite, quotient 0) (See General form below)

## Application in finance

The computation of compound interest leads to a geometric series:

When an initial amount A is deposited at an interest rate of p percent per time period then the value An of the deposit after n time-periods is given by

$A_n = A \left( 1 + {p\over100} \right)^n$

i.e., the values A=A0, A1, A2, A3, ... form a geometric sequence with quotient q = 1+(p/100).

## Mathematical notation

A finite sequence

$a_1,a_2,\dots,a_n = \{ a_i \mid i=1,\dots,n \} = \{ a_i \}_{i=1,\dots,n}$

or an infinite sequence

$a_0,a_1,a_2,\dots = \{ a_i \mid i\in\mathbb N \} = \{ a_i \}_{i\in\mathbb N}$

is called geometric sequence if

${ a_{i+1} \over a_i } = q$

for all indices i where q is a number independent of i. (The indices need not start at 0 or 1.)

### General form

Thus, the elements of a geometric sequence can be written as

$a_i = a_1 q^{i-1}$

Remark: This form includes two cases not covered by the initial definition depending on the quotient:

• a1 = 0 , q arbitrary: 0, 0•q = 0, 0, 0, ...
• q = 0 : a1, 0•a1 = 0, 0, 0, ...

(The initial definition does not cover these two cases because there is no division by 0.)

### Sum

The sum (of the elements) of a finite geometric sequence is

$a_1 + a_2 +\cdots+ a_n = \sum_{i=1}^n a_i$
$= a_1 ( 1+q+q^2+ \cdots +q^{n-1} ) = \begin{cases} a_1 { 1-q^n \over 1-q } & q \ne 1 \\ a_1 \cdot n & q = 1 \end{cases}$

The sum of an infinite geometric sequence is a geometric series:

$\sum_{i=0}^\infty a_0 q^i = a_0 { 1 \over 1-q } \qquad (\textrm {for}\ |q|<1)$