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If the arithmetic and geometric means of $a$ and $b$ be $A$ and $G$ respectively, then the value of $A-G$ will be
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Verified Answer
The correct answer is:
$\left[\frac{\sqrt{a}-\sqrt{b}}{\sqrt{2}}\right]^2$
Arithmetic mean of $a$ and $b=A=\frac{a+b}{2}$
and geometric mean $G=\sqrt{a b}$
Then, $A-G=\frac{a+b}{2}-\sqrt{a b}=$ $=\frac{a+b-2 \sqrt{a b}}{2}$
$=\frac{(\sqrt{a})^2+(\sqrt{b})^2-2(\sqrt{a})(\sqrt{b})}{2}=\left[\frac{\sqrt{a}-\sqrt{b}}{\sqrt{2}}\right]^2$
and geometric mean $G=\sqrt{a b}$
Then, $A-G=\frac{a+b}{2}-\sqrt{a b}=$ $=\frac{a+b-2 \sqrt{a b}}{2}$
$=\frac{(\sqrt{a})^2+(\sqrt{b})^2-2(\sqrt{a})(\sqrt{b})}{2}=\left[\frac{\sqrt{a}-\sqrt{b}}{\sqrt{2}}\right]^2$
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