Sequence And Series Question 251
Question: If $ ,a,\ b,,c $ be in G.P. and $ a+x,\ b+x,\ c+x $ in H.P., then the value of $ x $ is ( $ a,\ b,\ c $ are distinct numbers)
Options:
A) $ c $
B) $ b $
C) $ a $
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
$ a+x,\ b+x,\ c+x $ are in H.P.
$ \Rightarrow $ $ b+x=\frac{2(a+x)(c+x)}{(a+x)+(c+x)} $
$ \Rightarrow $ $ (b+x)(a+c+2x)=2(a+x)(c+x) $
$ \Rightarrow $ $ (a+c+2b)x+2x^{2}+ab+bc=2ac+2x(a+c)+2x^{2} $
$ \Rightarrow $ $ x(c+a-2b)=bc+ab-2ac $
$ \Rightarrow $ $ x(c+a-2b)=bc+ab-2b^{2} $ ( $ \because \ a,\ b,\ c $ are in G.P.)
Þ $ x(c+a-2b)=b(c+a-2b) $
Þ $ x=b $ , if $ c+a-2b\ne 0 $ .
BETA
