Sequence And Series Question 307
Question: If A and G are arithmetic and geometric means and $ x^{2}-2Ax+G^{2}=0 $ , then
[UPSEAT 2001]
Options:
A) $ A=G $
B) $ A \gt G $
C) $ A \lt G $
D) $ A=-G $
Show Answer
Answer:
Correct Answer: B
Solution:
$ x^{2}-2Ax+G^{2}=0 $ ?..(i) Let $ a,b $ are the two numbers whose A.M. is A and G.M. is G.
$ \therefore $ $ A=(a+b)/2,G^{2}=ab $ From (i), $ x^{2}-(a+b)x+ab=0 $
Þ $ x^{2}-ax-bx+ab=0 $
$ \Rightarrow x(x-a)-b(x-a)=0 $
Þ $ (x-a)(x-b)=0 $
$ \therefore a,b $ are the roots of the equation
$ \therefore $ $ A-G=\frac{a+b}{2}-\sqrt{ab} $ $ =\frac{1}{2}[(a+b)-2\sqrt{ab}] $ $ =\frac{1}{2}{{(\sqrt{a}-\sqrt{b})}^{2}}>0\Rightarrow $ $ A>G $ .
BETA
