if A be the AM and G be the GM between two numbers, show that the numbers are given by A + or -  sqrt (A^2 - G^2).

Dear Student,
If A and G are A.M. and G.M. between two positive numbers. Let these two positive numbers be a and b

∴ AM=a+b2-(1)
GM=ab -(2)

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(A2-G2)

(a – b)2 = 4 (A + G) (A – G)

(a-b)= 2(A+G)(A-G)   -(5)

From (3) and (5), we obtain
2a=2A +2(A+G)A-Ga=A+A+GA-G

Substituting the value of a in (3), we obtain

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



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