# 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 )] .

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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 )] .

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Solution:

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 (A2G2)

(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 .