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:

\( {G^{2}}=H \times A \)