Sequence And Series Question 215
Question: If three unequal non-zero real numbers $ a,\ b,\ c $ are in G.P. and $ b-c,\ c-a,\ a-b $ are in H.P., then the value of $ a+b+c $ is independent of
Options:
A) $ a $
B) $ b $
C) $ c $
D) None of these
Show Answer
Answer:
Correct Answer: D
Solution:
As given $ b^{2}=ac $ and $ \frac{2}{c-a}=\frac{1}{b-c}+\frac{1}{a-b} $
$ \Rightarrow $ $ 2(b-c)(a-b)=-{{(a-c)}^{2}} $
$ \Rightarrow $ $ 2(ab-ac-b^{2}+bc)=-{{{ (\sqrt{a}+\sqrt{c})(\sqrt{a}-\sqrt{c}) }}^{2}} $
$ \Rightarrow $ $ 2(ab-2b^{2}+bc)=-{{(\sqrt{a}-\sqrt{c})}^{2}}{{(\sqrt{a}+\sqrt{c})}^{2}} $
$ \Rightarrow $ $ 2b{{(\sqrt{a}-\sqrt{c})}^{2}}=-{{(\sqrt{a}-\sqrt{c})}^{2}}{{(\sqrt{a}+\sqrt{c})}^{2}} $
Þ $ 2b=-(a+c+2\sqrt{ac}),\ (\because \ \sqrt{a}-\sqrt{c}\ne 0) $ = $ -(a+c+2b) $
$ \Rightarrow $ $ a+b+c=-3b=-3\sqrt{ac} $ is not independent of $ a,\ b $ and $ c $ .
BETA
