Question

If the arithmetic, geometric and harmonic means between two positive real numbers be $A,G$ and $H$, then

• A. $A^2=GH$
• B. $H^2=AG$
• C. $G=AH$
• D. $G^2=AH$

Answer 1

The correct option is D

$G^2=AH$

Explanation for the correct option:

Given, $A,G$ and $H$ are arithmetic, geometric and harmonic means between two positive real numbers.

Then by definitions, we have:

If $A$ is an arithmetic mean between $a\&b$ then $A=\frac{a+b}{2}$.
If $G$ is an geometric mean between $a\&b$ then $G=\sqrt{ab}$.
If $H$ is an harmonic mean between $a\&b$ then $H=\frac{2ab}{a+b}$.
$\begin{array}{rcl}AH& =& \left(\frac{a+b}{2}\right)\left(\frac{2ab}{a+b}\right)\\ & =& ab\\ & =& G^2\\ AH& =& G^2\end{array}$

Hence, option (D) is correct.