Sequence And Series Question 341
Question: If $ A_1,\ A_2;G_1,\ G_2 $ and $ H_1,\ H_2 $ be $ AM’s,\ GM’s $ and $ HM’s $ between two quantities, then the value of $ \frac{G_1G_2}{H_1H_2} $ is
[Roorkee 1983; AMU 2000]
Options:
A) $ \frac{A_1+A_2}{H_1+H_2} $
B) $ \frac{A_1-A_2}{H_1+H_2} $
C) $ \frac{A_1+A_2}{H_1-H_2} $
D) $ \frac{A_1-A_2}{H_1-H_2} $
Show Answer
Answer:
Correct Answer: A
Solution:
Let the two quantities be $ a $ and $ b $ . Then $ a,\ A_1,\ A_2,\ b $ are in A.P. \ $ A_1-a=b-A_2\Rightarrow A_1+A_2=a+b $ ……(i) Again $ a,\ G_1,\ G_2,\ b $ are in G.P.
$ \therefore $ $ \frac{G_1}{a}=\frac{b}{G_2}\Rightarrow G_1G_2=ab $ …..(ii) Also $ a,\ H_1,\ H_2,\ b $ are in H.P.
$ \therefore $ $ \frac{1}{H_1}-\frac{1}{a}=\frac{1}{b}-\frac{1}{H_2}\Rightarrow \frac{1}{H_1}+\frac{1}{H_2}=\frac{1}{a}+\frac{1}{b} $
$ \Rightarrow $ $ \frac{H_1+H_2}{H_1H_2}=\frac{a+b}{ab}=\frac{A_1+A_2}{G_1G_2} $ [By (i) and (ii)]
$ \therefore $ $ \frac{G_1G_2}{H_1H_2}=\frac{A_1+A_2}{H_1+H_2} $ .
BETA
