Sequence And Series Question 194
Question: If the arithmetic and geometric means of a and b be $ A $ and $ G $ respectively, then the value of $ A-G $ will be
Options:
A) $ \frac{a-b}{a} $
B) $ \frac{a+b}{2} $
C) $ {{[ \frac{\sqrt{a}-\sqrt{b}}{\sqrt{2}} ]}^{2}} $
D) $ \frac{2ab}{a+b} $
Show Answer
Answer:
Correct Answer: C
Solution:
Arithmetic mean of $ a $ and $ b=A=\frac{a+b}{2} $ and geometric mean $ G=\sqrt{ab} $ Then $ A-G=\frac{a+b}{2}-\sqrt{ab} $ $ =\frac{a+b-2\sqrt{ab}}{2} $ $ =\frac{{{(\sqrt{a})}^{2}}+{{(\sqrt{b})}^{2}}-2(\sqrt{a})(\sqrt{b})}{2}={{[ \frac{\sqrt{a}-\sqrt{b}}{\sqrt{2}} ]}^{2}} $
BETA
