. If 1, 2, 3 be the arithmetic means and 1, 2, 3 be the harmonic means between  and , prove that 13 = 22 = 31 = .

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The arithmetic and harmonic mean of a series are equal only when the terms of each series are the same and can be represented by a. That is, the arithmetic series is a, a, a, ... and the harmonic series is 1/a, 1/a, 1/a, ... In each case the mean is a. So, if 1 is the arithmetic mean and the harmonic mean, a=1; if they are both 2, a=2; and if they are both 3, a=3.

This is easy to prove: arithmetic mean=(a+a+a+..)/n that is, na/n=a.

Harmonic mean: n/(1/a+1/a+1/a+...)=n/(n/a)=a.

The equivalence statement 13=22=31 makes no sense and requires explanation. The digits in this equivalence are limited to 1, 2 and 3, and 1 and 3, 2 and 2, and 3 and 1 have an arithmetic mean of 2, but their harmonic means differ. Also, 22 is the arithmetic mean of all three numbers and of 13 and 31, but not the harmonic mean.

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