Question

If $A_1{A}_2;G_1{G}_2;H_1{H}_2$ be two A.M.s, G.M.s and H.M.s between two numbers then prove: $\frac{G_1{G}_2}{H_1{H}_2}=\frac{A_1+A_2}{H_1+H_2}$

Answer 1

Let two nos be $a$ and$b$

$a,A_1,A_2,b$ are in A.P

$\frac{a+b}{2}=\frac{A_1+A_2}{2}$

$a+b=A_1+A_2$

$a,C_{r_1},b,C_{r_2}$ are in G.P

$\sqrt{ab}=\sqrt{C_{r_1}{C}_{r_2}}$

$ab=C_{r_1}{C}_{r_2}$

$a,H_1,H_2,b$ are in H.P

$\frac{1}{a},\frac{1}{H_1},\frac{1}{H_2},\frac{1}{b}$ are in A.P

$\frac{\frac{1}{a}+\frac{1}{b}}{2}=\frac{\frac{1}{H_1}+\frac{1}{H_2}}{2}$

$\frac{a+b}{ab}=\frac{H_1+H_2}{H_1{H}_2}$

$\frac{H_1{H}_2}{H_1+H_2}=\frac{ab}{a+b}$

$\frac{A_1+A_2}{H_1+H_2}\ast \frac{H_1{H}_2}{C_{r_1}{C}_{r_2}}=a+b\ast \frac{ab}{a+b}\ast \frac{1}{ab}$