Let a and b be the numbers such that A, G are A.M. and G.M. respectively between them. A.M. of a and b = $\frac{a+b}{2}$ i.e., A = $\frac{a+b}{2}$ ⇒ a + b = 2A ... (i) Also, G.M. of a and b = $\sqrt{ab}$ i.e., G = $\sqrt{ab}$ ⇒ $G^2$ = ab .....(ii) We know that, $(a - b)^2$ = $(a + b)^2$ – 4ab ⇒ $(a - b)^2$ = $(2A)^2 - 4G^2$ [By using (i) and (ii)] ⇒ $(a - b)^2$ = $4A^2 - 4G^2$ ⇒ a - b = 2 $\sqrt{A^2 - G^2}$ ... (iii) Adding (i) & (iii), we get 2a = 2A + $2\sqrt{A^2 - G^2}$ ⇒ a = A + $\sqrt{(A-G)(A+G)}$ Subtracting (iii) from (i), we get 2b = 2A - 2 $\sqrt{A^2 - G^2}$ ⇒ b = A - $\sqrt{(A-G)(A+G)}$ Hence, the numbers are A ± $\sqrt{(A-G)(A+G)}$