Question

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

Answer 1

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

Let, two positive numbers are a and b.

A= a+b 2 a+b=2A (1)

G= ab ab= ( G ) 2 (2)

We know that,

( a−b ) 2 = ( a+b ) 2 −4ab

Substitute the value of a+b and ab in above equation, we get

( a−b ) 2 =4 A 2 −4 G 2 =4( A 2 − G 2 ) =4( A−G )( A+G ) ( a−b )=2 ( A−G )( A+G ) (3)

Adding equation (1) and (3), we get

2a=2A+2 ( A−G )( A+G ) a=A+ ( A−G )( A+G )

Substituting the value of a in equation (1), we get

b=2A−A− ( A−G )( A+G ) b=A− ( A−G )( A+G )

Thus, the two numbers are A± ( A−G )( A+G ) .