If the arithmetic,geometric,and harmonic mean | vedclass

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Question

(A) For any two distinct positive real numbers $a$ and $b$,the arithmetic mean $A = \frac{a+b}{2}$,geometric mean $G = \sqrt{ab}$,and harmonic mean $H

Vedclass · Accepted Answer

$A > G > H$

Vedclass · Answer

(A) For any two distinct positive real numbers $a$ and $b$,the arithmetic mean $A = \frac{a+b}{2}$,geometric mean $G = \sqrt{ab}$,and harmonic mean $H = \frac{2ab}{a+b}$.We know that $A 	imes H = \left(\frac{a+b}{2}ight) 	imes \left(\frac{2ab}{a+b}ight) = ab = G^2$.Thus,$G^2 = AH$,which implies $G$ is the geometric mean of $A$ and $H$.For distinct positive real numbers,$A > G > H$ always holds true.

If the arithmetic,geometric,and harmonic means between two distinct positive real numbers are $A, G,$ and $H$ respectively,then the relation between them is

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