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 .

Answer :

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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