Geometric sequence/Citable Version

From Citizendium
< Geometric sequence
Revision as of 16:56, 14 May 2010 by imported>D. Matt Innis (replace with Approved version)
Jump to navigation Jump to search
This article has a Citable Version.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This version approved either by the Approvals Committee, or an Editor from the listed workgroup. The Mathematics Workgroup is responsible for this citable version. While we have done conscientious work, we cannot guarantee that this version is wholly free of mistakes. See here (not History) for authorship.
Help improve this work further on the editable Main Article!

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

  • (finite, length 6: 6 elements, quotient 2)
  • (finite, length 4: 4 elements, quotient −2)
  • (infinite, quotient )
  • (infinite, quotient 1)
  • (infinite, quotient −1)
  • (infinite, quotient 2)
  • (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

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

Mathematical notation

A finite sequence

or an infinite sequence

is called geometric sequence if

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

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

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