Question

If $A_1,A_2$ be two arithmetic means and $G_1,G_2$ be two geometric means between two positive numbers a and b, then $\frac{A_1+A_2}{G_1{G}_2}$

• A. $\frac{ab}{a+b}$
• B. $\frac{2ab}{a+b}$
• C. $\frac{a+b}{2ab}$
• D. $\frac{a+b}{ab}$

Answer 1

The correct option is D $\frac{a+b}{ab}$

For convinience lets say $a=p^3,b=q^3$
Then $G1=p^{2}q$
$G2=p{q}^2$
$G1G2=p^3\ast q^3=ab$
and $b=a+3d$
$d=(b-a)/3$
$A1=a+(b-a)/3=2a/3+b/3$
$A2=a+2d=a+(b-a)2/3=a/3+2b/3$
$A1+A2=a+b$
$(A1+A2)/\left(G1G2\right)=(a+b)/ab$