Sequence And Series Question 206
Question: If $ a,\ b,\ c $ are in G.P. and $ x,,y $ are the arithmetic means between $ a,\ b $ and $ b,\ c $ respectively, then $ \frac{a}{x}+\frac{c}{y} $ is equal to Roorkee 1969]
Options:
A) 0
B) 1
C) 2
D) $ \frac{1}{2} $
Show Answer
Answer:
Correct Answer: C
Solution:
Given that $ a,\ b,\ c $ are in G.P. So, $ b^{2}=ac $ ?..(i) $ x=\frac{a+b}{2} $ ?..(ii) $ y=\frac{b+c}{2} $ ?..(iii) Now $ \frac{a}{x}+\frac{c}{y}=\frac{2a}{a+b}+\frac{2c}{b+c}=\frac{2(ab+bc+2ca)}{ab+ac+b^{2}+bc} $ $ =\frac{2(ab+bc+2ca)}{(ab+ac+ac+bc)}=2 $ , $ { \because \ b^{2}=ac } $ . Trick: Let $ a=1,\ b=2,\ c=4, $ then obviously $ x=\frac{3}{2} $ and $ y=3 $ , then $ \frac{1}{3/2}+\frac{4}{3}=2 $ .
BETA
