Arithmetic and geometric means

In the following I'll consider sets of positive real numbers a₁, ..., a_N, N a positive integer. Arithmetic mean of the given numbers is defined as

whereas their geometric mean is given by

The two quantities always relate in the following manner known as the Arithmetic Mean - Geometric Mean Inequality (AM-GM, for short),

Here I am not going to prove the well known inequality but just emphasize a fact that was used by Cauchy in his proof. Namely, if the inequality holds for all N = 2ⁿ then it holds for all N ≥ 1. This would afford another example of a general proposition implied by its special case.

Thus, assume the inequality holds for all N = 2ⁿ and let N = 2ⁿ + m, where 0 < m < 2ⁿ and n > 0. For i = N+1, ..., 2ⁿ⁺¹, define the "missing" a's as

Since the inequality holds for N = 2ⁿ⁺¹ we have

Substituting a_i = (a₁ + ... + a_N)/N for i = N+1, ..., 2ⁿ⁺¹ results in

Adding similar terms on the left we get

which actually says that the arithmetic mean has not been changed by addition of new terms.

Dividing by the rightmost term and with one more step to go

or

Now raising both sides to the power of 2ⁿ⁺¹/N we finally get

Q.E.D.

There is a way to derive a complete proof of the inequality from the Pythagorean Theorem.

Related material Read more...
	The Means
	Averages, Arithmetic and Harmonic Means
	Expectation
	The Size of a Class: Two Viewpoints
	Averages of divisors of a given integer
	Family Statistics: an Interactive Gadget
	Averages in a sequence
	Geometric Meaning of the Geometric Mean
	A Mathematical Rabbit out of an Algebraic Hat
	AM-GM Inequality
	The Mean Property of the Mean
	Harmonic Mean in Geometry

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