If A, G, and H are the arithmetic mean,geomet | vedclass

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

Vedclass · Accepted Answer

$A > G > H$

Vedclass · Answer

(C) For any two positive real numbers $a$ and $b$,the arithmetic mean $A = \frac{a+b}{2}$,the geometric mean $G = \sqrt{ab}$,and the harmonic mean $H = \frac{2ab}{a+b}$.We know that for any two positive numbers,$A \ge G \ge H$.Specifically,$A 	imes H = \frac{a+b}{2} 	imes \frac{2ab}{a+b} = ab = G^2$.Since $G^2 = AH$,it implies $G$ is the geometric mean of $A$ and $H$,which confirms $A > G > H$ for distinct numbers.

If $A, G,$ and $H$ are the arithmetic mean,geometric mean,and harmonic mean of two given numbers respectively,then which of the following is true?

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