If A and G be A.M. and G.M., respectively between two positive numbers, prove that the numbers are .

Given: A and G are A.M. and G.M. between two positive numbers.

Let the two numbers be a and b.

∴ AM = A = —1

GM = G = √ab —2

From eq-1 and eq-2, we get

a + b = 2A —3

ab = G² —4

Substituting the value of a and b from eq-3 and eq-4 in

(a – b)² = (a + b)² – 4ab, we get

(a – b)² = 4A² – 4G² = 4 (A²–G²)

(a – b)² = 4 (A + G) (A – G)

(a – b) = 2 —5

From eq-3 and eq-5, we get

2a = 2A + 2

⇒ a = A+2

Substituting the value of a in eq-3, we get

b = 2A – A - = A –

Thus, the two numbers are .