# 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=$\frac{a+b}{2}-\left(1\right)$
GM=

From (1) and (2), we obtain

a+b= 2A    -(3)

ab =

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 = $4{A}^{2}-4{G}^{2}=4\left({A}^{2}-{G}^{2}\right)$

(a – b)2 = 4 (A + G) (A – G)

From (3) and (5), we obtain

Substituting the value of a in (3), we obtain
$b=2A-A-\sqrt{\left(A+G\right)\left(A-G\right)}=A-\sqrt{\left(A+G\right)\left(A-G\right)}$

Thus, the two numbers are .

Regards

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