Question

If A, G, H are arithmetic, geometric and harmonic means between a and b respectively, then A, G, H are

• A. in G.P.
• B. in A.P.
• C. in H.P.
• D. Real numbers

Answer 1

The correct option is A in G.P.

If A is an arithmetic mean between a and b then $A=\frac{a+b}{2}$.

If G is an geometric mean between a and b then $G=\sqrt{ab}$.

If H is an harmonic mean between a and b then $H=\frac{2ab}{a+b}$.

Now consider AH

$AH=\left(\frac{a+b}{2}\right)\left(\frac{2ab}{a+b}\right)$

$AH=ab$

$AH=G^2$

Since $AH=G^2$, therefore, A, G and H are in G.P.

Hence option A is correct.