Sequence And Series Question 355
Question: The harmonic mean of two numbers is 4 and the arithmetic and geometric means satisfy the relation $ 2A+G^{2}=27 $ , the numbers are
[MNR 1987; UPSEAT 1999, 2000]
Options:
A) $ 6,,3 $
B) 5, 4
C) $ 5,\ -2.5 $
D) $ -3,\ 1 $
Show Answer
Answer:
Correct Answer: A
Solution:
Let numbers be $ x $ and $ y $ . Then $ A=\frac{1}{2}(x+y),\ \sqrt{xy}=G $ or $ G^{2}=xy $ and $ ( \frac{1}{b}+\frac{1}{c}-\frac{1}{a} )( \frac{1}{c}+\frac{1}{a}-\frac{1}{b} ) $ ,
$ \Rightarrow $ $ G^{2}=4A $ Also, $ =( \frac{3}{b}-\frac{2}{a} )( \frac{1}{b} )=\frac{3}{b^{2}}-\frac{2}{ab} $
$ \Rightarrow $ $ (\because \ a,\ b,\ c $ So, $ x+y=9,\ xy=18 $ Hence numbers are 6 and 3.
BETA
