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  4. If A, G, H be respectively the A M, G M, and H . M between two positive numbers if x A=y G=z H where x, y, z are non-zero positive real number then x, y, z are in

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Q. If $A, G, H$ be respectively the $A M, G M$, and $H . M$ between two positive numbers if $x A=y G=z H$ where $x, y, z$ are non-zero positive real number then $x, y, z$ are in

Sequences and Series

Solution:

Let number are $a$ and $b$ then
$A =\frac{ a + b }{2}, $
$ G =\sqrt{a b} \text { and } H =\frac{2 a b}{a+b} $
$x A =y G \Rightarrow \frac{x}{y}=\frac{2 \sqrt{a b}}{a+b}$
also $y G=z H \Rightarrow \frac{y}{z}=\frac{2 \sqrt{a b}}{a+b} $
$ \Rightarrow \frac{x}{y}=\frac{y}{z}$ Hence vr $x, y, z$ are in GP