Under what conditions will the arithmetic mean...?

Question

Under what conditions will the arithmetic mean, (a+b)/2, of 2 nos. 'a' and 'b' equal to the nos. themselves?

How about the inequality involving the arithmetic and geometric mean of two nos. 'a' and 'b' assume the equality (a+b)/2 = square root of ab?

Answer 1

whwn a=b

Answer 2

The arithmetic mean of 'a' and 'b' will be equal to the nos. themselves if a = b. For example the arithmetic mean of 3 and 3 is equal to (3+3)/2 = 3.

The same goes for the inequality involing the arithmetic and geometric means. The (a+b)/2 = square root of ab is true when a = b.

Here is your proof:
(a+b)/2 = square root of (ab) -> square both sides of the equation
(a+b)/2 * (a+b)/2 = (ab)
(a^2 + 2ab + b^2)/4 = ab
a^2 + 2ab + b^2 = 4ab
a^2 - 2ab + b^2 = 0
(a-b)^2 = 0
a - b = 0
a = b

Answer 3

If and only if a = b.