Question

If A and G be A.M. and G.M., respectively between two positive numbers, prove that the numbers are A ± sqrt[(A +G )( A−G )] .

Answer 1

It is given that A and G are A.M. and G.M. between two positive numbers. Let these two positive numbers be a and b.

∴ 

From (1) and (2), we obtain

a + b = 2A … (3)

ab = G₂ … (4)

Substituting the value of a and b from (3) and (4) in the identity (a – b)² = (a + b)² – 4ab, we obtain

(a – b)² = 4A₂ – 4G₂ = 4 (A₂–G₂)

(a – b)² = 4 (A + G) (A – G)

From (3) and (5), we obtain

Substituting the value of a in (3), we obtain

Thus, the two numbers are .