Editing Geometric mean (section)

===Iterative means===
The geometric mean of a data set [[inequality of arithmetic and geometric means|is less than]] the data set's [[arithmetic mean]] unless all members of the data set are equal, in which case the geometric and arithmetic means are equal. This allows the definition of the [[arithmetic-geometric mean]], an intersection of the two which always lies in between.

The geometric mean is also the '''arithmetic-harmonic mean''' in the sense that if two [[sequence]]s (<math display="inline">a_n</math>) and (<math display="inline">h_n</math>) are defined:
:<math>a_{n+1} = \frac{a_n + h_n}{2}, \quad a_0 = x</math>

and
:<math>h_{n+1} = \frac{2{a_n}{h_n}}{a_n + h_n}, \quad h_0 = y</math>

where <math display="inline">h_{n+1}</math> is the [[harmonic mean]] of the previous values of the two sequences, then <math display="inline">a_n</math> and <math display="inline">h_n</math> will converge to the geometric mean of <math display="inline">x</math> and <math display="inline">y</math>. The sequences converge to a common limit, and the geometric mean is preserved:
:<math>\sqrt{a_{i+1} h_{i+1}} =
  \sqrt{\frac{a_i + h_i}{2}\frac{2{a_i}{h_i}}{a_i + h_i}} =
  \sqrt{{a_i}{h_i}}
</math>

Replacing the arithmetic and harmonic mean by a pair of [[generalized mean]]s of opposite, finite exponents yields the same result.