Question

If $A$ and $G$ be A.M. and G.M. of two given positive real numbers $a$ and $b$ respectively, then $A$ and $G$ are related as

• A. $A\ge G$
• B. $G\ge A$
• C. $A=G$
• D. $A=-G$

Answer 1

Answer: (A)

Let $A$ and $G$ be A.M and G.M. of two given positive real numbers $a$ and $b$ respectively.
Then, $A=\frac{a+b}{2}$ and $G=\sqrt{ab}$
Thus, we have
$A-G=\frac{a+b}{2}-\sqrt{ab}=\frac{a+b-2\sqrt{ab}}{2}$
$=\frac{(\sqrt{a}-\sqrt{b}{)}^2}{2}\ge 0$
Hence, $A-G\ge 0\Rightarrow A\ge G$