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).