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

Let the two positive numbers be a and b.

Therefore, A = ⇒ a + b = 2A

and G = ⇒ ab = G2

The equation whose roots are a and b is

x2 – (a + b)x + ab = 0

⇒x2 – 2Ax + G2 = 0 …….(i)

or x =

= A ±

= A ±

The roots of (i) are a and b,

Hence, the two positive number are given by

A ±

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