geometric sequence

A sequence of the form

$$a,ar,a{r}^2,a{r}^3,\mathrm{\dots }$$

of real or complex numbers is called geometric sequence.  of the geometric sequence is thus that every two consecutive members of the sequence have the constant ratio $r$, called usually the common ratio of the sequence (if  $ar=0$, speaking the ratio of members does not exist).

The $n^{\mathrm{th}}$ member of the geometric sequence has the

$$a_n=a{r}^{n-1}.$$

Let  $a\ne 0$.  The sequence is convergent for    having the limit (http://planetmath.org/LimitOfRealNumberSequence) 0, and for  $r=1$  having as constant sequence the limit $a$.

When the members of the sequence are positive numbers, each member is the geometric mean of the preceding and the following member; the name “geometric sequence”(or “geometric series”) is due to this fact (a fact is true for the harmonic series and harmonic mean).

Title	geometric sequence
Canonical name	GeometricSequence
Date of creation	2013-03-22 14:38:52
Last modified on	2013-03-22 14:38:52
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	14
Author	pahio (2872)
Entry type	Definition
Classification	msc 40-00
Related topic	GeometricSeries
Related topic	LimitOfRealNumberSequence
Defines	common ratio