What is the Relation between geometric mean and arithmetic mean?

Answer 1

The mean of the numbers a1, a2, a3, ..., an is equal to (a1 + a2 + a3 +... + an)/n. This number is also called the average or the arithmetic mean.

The geometric mean of the positive numbers a1, a2, a3, ... an is the n-th roots of [(a1)(a2)(a3)...(an)]

Given two positive numbers a and b, suppose that a< b. The arithmetic mean, m, is then equal to (1/2)(a + b), and, a, m, b is an arithmetic sequence. The geometric mean, g, is the square root of ab, and, a, g, b is a geometric sequence. For example, the arithmetic mean of 4 and 25 is 14.5 [(1/2)(4 + 25)], and arithmetic sequence is 4, 14.5, 25. The geometric mean of 4 and 25 is 10 (the square root of 100), and the geometric sequence is 4, 10, 25.

It is a theorem of elementary algebra that, for any positive numbers a1, a2, a3, ..., an, the arithmetic mean is greater than or equal to the geometric mean. That is:

(1/n)(a1, a2, a3, ..., an) ≥ n-th roots of [(a1)(a2)(a3)...(an)]