Browse Questions

# If arithmetic, geometric and harmonic means between two positive real numbers is A, G and H respectively, then, which of the following is true?

$\begin{array}{1 1} A>G>H \\G>A>H \\ A> G < H \\ H > G > A\end{array}$

Explanation : Let the two real numbers be a,b
$A=(a+b)/2\quad\;G=\sqrt{ab}\quad\;H=2a/(a+b)$
$A-G=(a+b)/2\;-\;\sqrt{ab}=(a+b-2\sqrt{ab})/2$
$=(\sqrt{a}-\sqrt{b})^2/2$
Which is always positive as it is a square.
Therefore A>G.