Sequence And Series Question 308
Question: If ln $ (a+c) $ , In $ (c-a) $ , In $ (a-2b+c) $ are in A.P., then
[IIT Screening 1994]
Options:
A) $ a,\ b,\ c $ are in A.P.
B) $ a^{2},\ b^{2},\ c^{2} $ are in A.P.
C) $ a,\ b,\ c $ are in G.P.
D) $ a,\ b,\ c $ are in H.P
Show Answer
Answer:
Correct Answer: D
Solution:
$ 2\ln (c-a)=\ln (a+c)+ln\ (a-2b+c) $
$ \Rightarrow $ $ {{(c-a)}^{2}}=(a+c)(a-2b+c) $
$ \Rightarrow $ $ c^{2}+a^{2}-2ac={{(a+c)}^{2}}-2b(a+c) $
$ \Rightarrow $ $ c^{2}+a^{2}-2ac=a^{2}+c^{2}+2ac-2ab-2bc $
$ \Rightarrow $ $ b(a+c)=2ac $
$ \Rightarrow $ $ b=\frac{2ac}{a+c} $ . Hence, $ a,\ b,\ c $ are in H.P.
BETA
