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# Given that four positive numbers are in HP, which of the following is true.

$\begin{array}{1 1} ab > cd \\ ac >bd \\ ad > bc \\ none\;of\;the\;above \end{array}$

Explanation :Sinc a,b,c,d are in HP
$HM of a\;\xi\;c\; is\; b$
$HM of b\;\xi\;d\;is\;c\;$
$GM of a\;\xi\;c\; is\;\sqrt{ac}$
$GM of b\;\xi\;d\; is\;\sqrt{bd}$
Knowing that GM>HM for any two numbers
$\sqrt{ac}>\;b\qquad\sqrt{bd}>c$
$ac\;>\;b^2\;(1)\qquad\;bd\;>\;c^2\;(2)$
$Multiplying (1)\;\xi\;(2)$
$acbd\;>\;b^2 c^2\qquad\;(3)$
$Dividing \;(3)\; by \;bc$
$ad\;>bc\;.$