# geometric sequence

A sequence of the form

 $a,\,ar,\,ar^{2},\,ar^{3},\,\ldots$

of real or complex numbers is called .  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

 $a_{n}=ar^{n-1}.$

Let  $a\neq 0$.  The sequence is convergent for  $|r|<1$  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 GeometricSequence 2013-03-22 14:38:52 2013-03-22 14:38:52 pahio (2872) pahio (2872) 14 pahio (2872) Definition msc 40-00 GeometricSeries LimitOfRealNumberSequence common ratio