Sequence And Series Question 265
Question: If $ a,\ b,\ c $ are in H.P., then for all $ n\in N $ the true statement is
[RPET 1995]
Options:
A) $ a^{n}+c^{n}<2b^{n} $
B) $ a^{n}+c^{n}>2b^{n} $
C) $ a^{n}+c^{n}=2b^{n} $
D) None of the above
Show Answer
Answer:
Correct Answer: B
Solution:
For two numbers $ a $ and $ c $ $ \frac{a^{n}+c^{n}}{2}>{{( \frac{a+c}{2} )}^{n}} $ (Where $ n\in N,\ n>1 $ ) $ \because $ $ A.M.>H.M. $
$ \therefore $ $ \frac{a+b}{2}>b $ $ (\because \ a,\ b,\ c $ are in A.P.)
$ \Rightarrow $ $ {{( \frac{a+c}{2} )}^{n}}>b^{n} $
$ \Rightarrow $ $ \frac{a^{n}+c^{n}}{2}>{{( \frac{a+c}{2} )}^{n}}> \frac{a^{n}+c^{n}}{2} $ .
BETA
