Harmonic Mean of Two Numbers
For the special case of just two numbers and, the harmonic mean can be written
In this special case, the harmonic mean is related to the arithmetic mean and the geometric mean by
So
meaning the two numbers' geometric mean equals the geometric mean of their arithmetic and harmonic means.
As noted above this relationship between the three Pythagorean means is not limited to n equals 1 or 2; there is a relationship for all n. However, for n = 1 all means are equal and for n = 2 we have the above relationship between the means. For arbitrary n ≥ 2 we may generalize this formula, as noted above, by interpreting the third equation for the harmonic mean differently. The generalized relationship was already explained above. If one carefully observes the third equation one will notice it also works for n = 1. That is, it predicts the equivalence between the harmonic and geometric means but it falls short by not predicting the equivalence between the harmonic and arithmetic means.
The general formula, which can be derived from the third formula for the harmonic mean by the reinterpretation as explained in relationship with other means, is
Notice that for n = 2 we have
where we used the fact that the arithmetic mean evaluates to the same number independent of the order of the terms. This equation can be reduced to the original equation if we reinterpret this result in terms of the operators themselves. If we do this we get the symbolic equation
because each function was evaluated at
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