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

It is given that $A$ and $G$ are $A . M .$ and $G . M .$ between two positive numbers.

Let these two positive numbers be $a$ and $b$

$\therefore A M=A=\frac{a+b}{2}$        .........$(1)$

$G M=G=\sqrt{a b}$       ........$(2)$

From $(1)$ and $(2),$ we obtain

$a+b=2 A$          ..........$(3)$

$a b=G^{2}$       ........$(4)$

Substituting the value of $a$ and $b$ from $(3)$ and $(4)$ in the identity

$(a-b)^{2}=(a+b)^{2}-4 a b$

We obtain

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

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

$(a-b)=2 \sqrt{(A+G)(A-G)}$         .........$(5)$

From $(3)$ and $(5),$ we obtain

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

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

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

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

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