Sequence And Series Question 199
Question: If $ \frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{a}+\frac{1}{c} $ , then $ a,\ b,\ c $ are in
[MNR 1984; MP PET 1997; UPSEAT 2000]
Options:
A) A.P.
B) G.P.
C) H.P.
D) In G.P. and H.P. both
Show Answer
Answer:
Correct Answer: C
Solution:
Since the reciprocals of $ a $ and $ c $ occur on RHS, let us first assume that $ a,\ b,\ c $ are in H.P. So that $ \frac{1}{a},\ \frac{1}{b},\ \frac{1}{c} $ are in A.P.
$ \Rightarrow $ $ \frac{1}{b}-\frac{1}{a}=\frac{1}{c}-\frac{1}{b}=d $ , say
$ \Rightarrow $ $ \frac{a-b}{ab}=d=\frac{b-c}{bc}\Rightarrow a-b=abd $ and $ b-c=bcd $ Now LHS $ =-\frac{1}{a-b}+\frac{1}{b-c}=-\frac{1}{abd}+\frac{1}{bcd} $ $ =\frac{1}{bd}( \frac{1}{c}-\frac{1}{a} )=\frac{1}{bd}(2d)\Rightarrow \frac{2}{b}=\frac{1}{a}+\frac{1}{c}= $ RHS
$ \therefore \ \ a,.\ b,\ c $ are in H.P. is verified.
BETA
