Q. 293.8( 64 Votes )

# 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

(ab)2 = (a + b)2 – 4ab, we get

(ab)2 = 4A2 – 4G2 = 4 (A2G2)

(ab)2 = 4 (A + G) (AG)

(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 .

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