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Question: an arithmetic mean of two numbers is always between the two numbers being averaged. Is this also true of harmonic means and geometric means?

Yes, it is. Let's take a look at both types of means, and see if we can figure out why. In each case we'll make the assumption that we're taking the mean of two positive, distinct numbers. Positive, because goeometric means and harmonic means are generally defined to be over the set of positive reals, and distinct, because having non-distinct numbers makes the mean equal to the original number, which isn't terribly interesting.

Geometric Mean

The geometric mean is the square root of the product of the two numbers. So if the numbers are a and b, then the geometric mean is SQR{ab}.

We will say, without loss of generality, that the two numbers are a and b, with a < b.

Multiply both sides of the equation by b:

ab < b2

Taking the square root of both sides gives SQR{ab} < b

Similarly, if we multiply the first equation by a, we get:

a2 < ab, or a < SQR{ab}

Therefore, a < geometric mean < b.

Harmonic Mean

The harmonic mean is the reciprocal of the average of the reciprocals, or 2ab/(a + b). Again, we'll assume, without loss of generality that a < b.

Multiply both sides of this equation by a, and then add ab:

a2 + ab < 2ab

Now factor the left side:

a(a + b) < 2ab.

Finally, divide both sides by (a + b):

a < 2ab/(a + b).

We can repeat the process on the other side:

a < b
ab < b2
ab + ab < b2 + ab
2ab < b(a + b)
2ab/(a + b) < b

Therefore, a < harmonic mean < b

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