Question

For nonnegative real numbers a1, an, the arithmetic mean or average is defined by a1 +.+an n and the geometric mean is defined by Vai..an. In this problem, we will prove the AM-GM inequality. More precisely, for all positive integers n>2, given any nonnegative real numbers a1, -, An, we will show that ai +...+an Vaj.-.An. n We will do so by induction on n but in an unusual way.

Answer 1

The AM-GM inequality establishes that for any set of nonnegative real numbers the arithmetic mean is at least as great as the geometric mean, and this can be proven using induction.

Step-by-step explanation:

The student is asking about the AM-GM inequality, a fundamental result in algebra that relates the arithmetic mean and the geometric mean of a set of nonnegative real numbers. The arithmetic mean is the sum of values divided by the number of values, and the geometric mean is the nth root of the product of those values. To prove the AM-GM inequality, we assume it holds for a set of numbers, and then by induction extend it to show that for any set of n nonnegative real numbers a1, a2, ..., an, the arithmetic mean is always greater than or equal to the geometric mean: (a1 + a2 + ... + an) / n ≥ √(a1 * a2 * ... * an).