A sequence of the form $$a,\,ar,\,ar^2,\,ar^3,\,\ldots$$ of real or complex numbers is called geometric sequence. Characteristic 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 $ar = 0$ strictly speaking the ratio of members does not exist).

The $n^\mathrm{th}$ member of the geometric sequence has the formula $$a_n = ar^{n-1}.$$ Let $a \neq 0$ The sequence is convergent for $|r| < 1$ , having the limit 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 comparable fact is true for the harmonic series and harmonic mean).