We recall that the arithmetic mean of a₁, a₂, ..., a_n is

and the geometric mean is

We restrict a₁, ..., a_n to be positive in discussing geometric means, since otherwise the definition might express the mean as an even root of a negative number.

We shall use A_n for the arithmetic mean of a₁, a₂, ..., a_n and G_n for the geometric mean.

THEOREM: Let a₁, a₂, ..., a_n be positive real numbers. Then

that is,
 

Proof: We proceed by mathematical induction. When n = 1, it is clear that A₁ = a₁ and G₁ = a₁. Hence A₁ = G₁, and the theorem holds for n = 1.

We next prove it for n = 2. Since a₁ and a₂ are positive,  and  exist in the real number system by the roots property of Section 10.1, and so

by Problem 3(b) of section 10.1. It follows that

This is precisely the statement  of the theorem for n = 2.
 
 

Monday, June 22, 1998