Sequence And Series Question 107
Question: If a, b, c and d are in H.P. then
Options:
A) $ a^{2}+c^{2}>b^{2}+d^{2} $
B) $ a^{2}+d^{2}>b^{2}+c^{2} $
C) $ ac+bd>b^{2}+c^{2} $
D) $ ac+bd>b^{2}+d^{2} $
Show Answer
Answer:
Correct Answer: C
Solution:
[c] If a, b, c and d are in H.P., then $ \frac{1}{a};\frac{1}{b},\frac{1}{c} $ and $ \frac{1}{d} $ will be in A.P. Therefore, $ \frac{1}{b}-\frac{1}{a}=\frac{1}{c}-\frac{1}{b}=\frac{1}{d}-\frac{1}{c} $
$ \Rightarrow b=\frac{2ac}{a+c} $ G.M. between a and c= $ \sqrt{ac} $ Now, since G.M.>H.M., we have $ \sqrt{ac}>b $ Or $ ac>b^{2} $ …(1) Similarly, $ \sqrt{bd}>c $ Or $ bd>c^{2} $ …(2) Adding (1) and (2), we get $ ac+bd>b^{2}+c^{2} $
BETA
