Sequence And Series Question 286
Question: If $ a,\ b,\ c,\ d $ be 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:
$ a,\ b,\ c,\ d $ be in H.P., then $ \frac{1}{a},\ \frac{1}{b},\ \frac{1}{c},\ \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 as $ G.M>H.M $ ., so here $ \sqrt{ac}>b $ or $ ac>b^{2} $ . Similarly $ \sqrt{bd}>c $ or $ bd>c^{2} $ Adding, $ ac+bd>b^{2}+c^{2} $
BETA
