if arithmetic mean, geometric mean and harmonic mean between two numbers a and b are A, G & H then A, G H will be 

Explanation:

If A is the arithmetic mean between a and b, 

\(⇒ A=\frac{a+b}{2}\)

If G is the geometric mean between a and b,

⇒ G = √ab

If H is the harmonic mean between a and b,

\(⇒ H=\frac{2ab}{a+b}\)

Now, AH = \(\frac{a+b}{2}\times \frac{2ab}{a+b}\)

⇒ AH = ab

⇒ AH = G²

This is a form of a G.P.

Hence, If the arithmetic mean, geometric mean, and Harmonic mean between two numbers 'a' and 'b' are A, G, and H respectively, then A, G, H will be in geometric series.