5.0 Relation among A.M., G.M. and H.M.

5.1 Recognization of A.P., G.P. and H.P.

Let $a$ and $b$ be two real positive and unequal numbers and $A$, $G$, $H$ are arithmetic, geometric and harmonic mean respectively between them.

$$A = \frac{{a + b}}{2},{\text{ }}G = \sqrt {ab} ,{\text{ }}H = \frac{{2ab}}{{a + b}}$$

Now, $$\begin{equation} \begin{aligned} AH = \left( {\frac{{a + b}}{2}} \right)\left( {\frac{{2ab}}{{a + b}}} \right) = ab = {G^2} \\ \Rightarrow {G^2} = AH...(1) \\\end{aligned} \end{equation} $$

i.e., $G$ is the geometric mean between $A$ and $H$.

Again $$\begin{equation} \begin{aligned} A - G = \frac{{a + b}}{2} - \sqrt {ab} = \frac{{a + b - 2\sqrt {ab} }}{2} = {\left( {\frac{{\sqrt a - \sqrt b }}{{\sqrt 2 }}} \right)^2} > 0 \\ A - G > 0 \\ \Rightarrow A > G...(2) \\\end{aligned} \end{equation} $$

From equation $(1)$ and $(2)$, $$\begin{equation} \begin{aligned} \frac{G}{H} = \frac{A}{G} > 1 \\ \frac{G}{H} > 1 \\ G > H...(3) \\\end{aligned} \end{equation} $$

From equation $(2)$ and $(3)$, $$A > G > H$$

Hence, $G$ lies between $A$ and $H$.