Sequence And Series Question 221
Question: If $ A_1,\ A_2 $ are the two A.M.’s between two numbers $ a $ and $ b $ and $ G_1,\ G_2 $ be two G.M.’s between same two numbers, then $ \frac{A_1+A_2}{G_1.G_2}= $
[Roorkee 1983; DCE 1998]
Options:
A) $ \frac{a+b}{ab} $
B) $ \frac{a+b}{2ab} $
C) $ \frac{2ab}{a+b} $
D) $ \frac{ab}{a+b} $
Show Answer
Answer:
Correct Answer: A
Solution:
Given that $ a,\ A_1,\ A_2,\ b $ are in A.P. Therefore $ A_1=\frac{a+A_2}{2},\ A_2=\frac{A_1+b}{2} $
$ \Rightarrow $ $ x+y+z=15 $
$ \Rightarrow $ $ =9+15+a=\frac{5}{2}(9+2) $ or $ A_1+A_2=a+b $ ?..(i) and $ a,\ G_1,\ G_2,\ b $ are in G.P. Therefore $ G_1^{2}=aG_2,\ G_2^{2}=bG_1 $ ?..(ii)
$ \Rightarrow G_1^{2}G_2^{2}=abG_1G_2\Rightarrow G_1G_2=ab $ Hence $ \frac{A_1+A_2}{G_1G_2}=\frac{a+b}{ab} $ Trick: Let $ a=1,\ b=2 $ , then $ A_1+A_2=1+2=3 $ and $ G_1\ .\ G_2=2\times 1=2 $
$ \therefore \ $ $ \frac{A_1+A_2}{G_1G_2}=\frac{3}{2} $ , which is given by (a).
BETA
