# If a, g and h are the AM, GM and HM respectively of two positive numbers x and y, then identify the correct statement.

$(a)\;h\;is\;the\;HM\;between\;a\;and\;g\qquad(b)\;a\;is\;the\;AM\;between\;h\;and\;g\qquad(c)\;g\;is\;the\;GM\;between\;a\;and\;h\qquad(d)\;No\;relationships\;exits\;between\;a,g\;and\;h$

Answer : (c) $g\;is\;the\;GM\;between\;a\;and\;h$
Explanation : By definition , $a=\frac{x+y}{2}\;,g=\sqrt{xy}\;and\;h=\frac{2xy}{x+y}$
$Multiplying\;a*h\;=\frac{x+y}{2}\;*\frac{2xy}{x+y}=xy=g^2$
$g^2=ah=\frac{a}{g}=\frac{g}{h}$
Therefore a,g,h are in GP and g is the GM between a and h.