Sequence And Series Question 25
Question: If $ a,\ b,\ c $ be in H.P., then
Options:
A) $ a^{2}+c^{2}>b^{2} $
B) $ a^{2}+b^{2}>2c^{2} $
C) $ a^{2}+c^{2}>2b^{2} $
D) $ a^{2}+b^{2}>c^{2} $
Show Answer
Answer:
Correct Answer: C
Solution:
From the section of inequality, we know that A.M. of $ n^{th} $ powers $ (n-1)=1,\ 2\ i.e.,\ n=2,\ 3 $ power of A.M. $ i.e. $ $ \frac{1}{2}(a^{n}+c^{n})>{{( \frac{1}{2}(a+c) )}^{n}} $ Considering two quantities $ b^{2}=ac $ and c or $ \frac{1}{2}(a^{n}+c^{n})>{{(A)}^{n}} $ or $ \frac{1}{2}(a^{n}+c^{n})>{{(H)}^{n}} $ Since A.M. >H.M. $ \frac{1}{2}(a^{n}+c^{n})>{{(b)}^{n}} $
$ \Rightarrow $ $ a^{n}+c^{n}>2b^{n} $ Putting $ n=2 $ , we have $ a^{2}+c^{2}>2b^{2} $ .
BETA
