Sequence And Series Question 295
Question: If $ A_1,\ A_2;G_1,\ G_2 $ and $ H_1,\ H_2 $ be two A.M.s, G.M.s and H.M.s between two numbers respectively, then $ \frac{G_1G_2}{H_1H_2}\times \frac{H_1+H_2}{A_1+A_2} $ =
[RPET 1997]
Options:
A) 1
B) 0
C) 2
D) 3
Show Answer
Answer:
Correct Answer: A
Solution:
Let $ a $ and $ b $ two numbers respectively. Sum of $ n $ A.M.’s $ =n\times $ single A.M.
$ \Rightarrow $ $ A_1+A_2=2\times ( \frac{a+b}{2} )=a+b $ Product of $ n $ G.M.’s = (Single G.M.)n
$ \Rightarrow $ $ G_1.G_2={{(\sqrt{ab})}^{2}}=ab $ $ \frac{1}{a},\ \frac{1}{H_1},\ \frac{1}{H_2},\ \frac{1}{b} $ are in A.P.
$ \Rightarrow $ $ \frac{1}{H_1}+\frac{1}{H_2}=\frac{1}{a}+\frac{1}{b}=\frac{a+b}{ab} $
$ \Rightarrow $ $ \frac{H_1H_2}{H_1+H_2}=\frac{G_1G_2}{A_1+A_2} $
$ \Rightarrow $ $ \frac{G_1G_2}{H_1H_2}\times \frac{H_1+H_2}{A_1+A_2}=1 $
BETA
