If A and G be A.M. and GM., respectively between two positive numbers, prove that the numbers are `A+-sqrt((A+G)(A-G))`.

Answer

N/a

Explanation

Let two number be a and b.
`:. A=(a+b)/(2) " and " G=sqrt(ab)`
`implies a+b=2A " and " G^(2)=ab`
`implies b=2A-a`
Now ` G^(2)=ab implies G^(2)=a(2A-a)`
`implies a^(2)-2Aa+G^(2)=0`
`a=(2Apmsqrt(4A^(2)-4G^(2)))/(2)`
`impliesa=Apmsqrt(A^(2)-G^(2))`
If `a=A+sqrt(A^(2)-G^(2)),`then b=2A-a
`implies b=2A-(A+sqrt(A^(2)-G^(2))=A-sqrt(A^(2)-G^(2))`
If `a=A-sqrt(A^(2)-G^(2))`, then b=2A-a
`implies n=2A-(A-sqrt(A^(2)-G^(2)))=A+sqrt(A^(2)-G^(2))`
Therefore, numbers are `A+sqrt(A^(2)-G^(2))` and `A-sqrt(A^(2)-G^(2)).`
`implies Apmsqrt((A+G)(A-G))" " ` Hence Proved.