(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

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

Class 11 Mathematics Sequences and Series Relation between A.P., G.P.

Medium

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?

A
$A > H > G$
B
$G > A > H$
C
$A > G > H$
D
$A < G < H$

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Relation between A.P., G.P.

Three non-zero real numbers form an $A.P.$ and the squares of these numbers taken in the same order form a $G.P.$ Then the number of all possible common ratios of the $G.P.$ is

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If $a_1, a_2, \dots, a_n$ are positive real numbers such that $a_1 \cdot a_2 \cdot \dots \cdot a_n = 1$,then their sum is:

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If $a, b, c$ are in $A.P.$,$b, c, d$ are in $G.P.$,and $c, d, e$ are in $H.P.$,then $a, c, e$ are in

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If $a$ and $b$ are two different positive real numbers,then which of the following relations is true?

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$A$ car completes the first half of its journey with a velocity $v_1$ and the rest half with a velocity $v_2$. Then the average velocity of the car for the whole journey is

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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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