Another mean, the geometric mean is defined for non-negative values only. For x1,...,xn, it is the n'th root of (x1*...*xn).

The AM-GM inequality relates the geometric mean to the more famous arithmetic mean (and also to the lesser known harmonic mean).

So what's so geometric about this mean?

Place two line segments, of lengths a and b, adjacent on the same line. Construct a circle with the line segment a+b as its diameter. Obviously, its radius is the arithmetic mean (a+b)/2. BUT...

Raise a perpendicular from between a and b to the circle. The length of this perpendicular is precisely sqrt(ab) -- the geometric mean of a and b!

In simple terms, the geometric means are the terms between two nonconsecutive terms in a geometric sequence - a sequence where each number is multiplied by a common ratio rather than related by a common difference.
The formula for a geometric sequence is:
an=a1rn-1
an is the nth or last term in the sequence, a1 is the first, r is the common ratio, and n is the number of the term - a5 is the fifth term and n=5, for example.
Finding a geometric mean: Input values for the first and nth term, value of n (the number of geometric means you want between the two + 2 to represent the first and nth terms), and solve for the common ratio. Now you can use the common ratio and a1 to find the means of that sequence.

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