Question

Show what relation exists between arithmetic mean, A, Geometric mean, G and Harmonic mean, H?

Answer 1

Suppose x and y be the two numbers.

Then, Arithmetic mean, A is given by:

$A=\frac{x+y}{2} ~~~~~Eq.I$

Geometric mean, G is given by:

$G=\sqrt{xy}$

Squaring on both sides:

$G^{2}=xy~~~~~Eq.II$

For two numbers x and y harmonic mean, H is given by:

$H=\frac{2xy}{x+y}$

This can be rewritten as:

$H={xy}\cdot \frac{{2}}{x+y}={xy}\cdot \frac{1}{\frac{x+y}{{2}}}$

Substitute $xy={G^{2}} ~~(from~Eq.II)$ and $\frac{x+y}{2} ={A}~~(from~Eq.I)$.

$H={G^{2}}\cdot \frac{1}{{A}}$

$or, ~~{G^{2}}=H \times A$

:

The following relation exists:

${G^{2}}=H \times A$