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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 ± square root of (A+G )(A−G ) .

29.   If A and G be A.M. and G.M., respectively between two positive numbers, prove that the numbers are A \pm \sqrt{( A+G)(A-G)}

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  If A and G be A.M. and G.M., respectively between two positive numbers,
Two numbers be a and b.

AM=A=\frac{a+b}{2}

\Rightarrow a+b=2A...................................................................1

GM=G=\sqrt{ab}

\Rightarrow ab=G^2...........................................................................2

We know (a-b)^2=(a+b)^2-4ab

Put values from equation 1 and 2,

(a-b)^2=4A^2-4G^2

(a-b)^2=4(A^2-G^2)

(a-b)^2=4(A+G)(A-G)

(a-b)=4\sqrt{(A+G)(A-G)}..................................................................3

From 1 and 3 , we have

2a=2A+2\sqrt{(A+G)(A-G)}

\Rightarrow a=A+\sqrt{(A+G)(A-G)}

Put value of a in equation 1, we get

b=2A-A-\sqrt{(A+G)(A-G)}

\Rightarrow b=A-\sqrt{(A+G)(A-G)}

Thus, numbers are A \pm \sqrt{( A+G)(A-G)}

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