Not able to solve it for n>2 observations

Question

The relation between $\textrm{AM, GM , HM}$ is given as follows $AM \times HM=( GM )^2$

Does it also follow if the number of observations are more than $2$?

Answer 1

You can try a numerical example for $n=3$:

$$\begin{array}{|c|c|} \hline i&1&2&3 \\ \hline x_i &1&2&3 \\ \hline \end{array}$$

$AM=2$

$HM=\frac3{\frac{1}{1}+\frac{1}{2}+\frac{1}{3}}=1.\overline{63}$

$GM=\sqrt[3]{1\cdot 2\cdot 3}\Rightarrow GM^2=6^{\frac23}\approx 3.30193$

It is sufficient to show that $AM\times HM\neq GM^2$ to conclude that the equation does not hold in general for $n>2$.