The geometric mean of a sequence 
is defined by
 |
(1)
|
Thus,
 |  |  |
(2)
|
 |  |  |
(3)
|
and so on.
For two positive lengths 
and 
, place 
, 
,
and 
in this order on a line with 
and 
, and draw a semicircle with
diameter 
. Let the perpendicular
to 
at 
meet the semicircle at 
. The right triangle 
is split into similar
triangles 
and 
,
giving 
,
so 
is the geometric mean (Joyce). Setting 
gives a geometric construction of the square
root of 
.
The geometric mean of a list of numbers may be computed using GeometricMean[list] in the Wolfram Language package DescriptiveStatistics` .
For 
,
the geometric mean is related to the arithmetic mean 
and harmonic
mean 
by
 |
(4)
|
(Havil 2003, p. 120).
The geometric mean is the special case 
of the power mean and is one
of the Pythagorean means.
Hoehn and Niven (1985) show that
 |
(5)
|
for any positive constant 
.
