Applications Of Derivatives Question 22
Question: The largest term in the sequence $ a_{n}=\frac{n^{2}}{n^{3}+200} $ is given by
Options:
A) $ \frac{529}{49} $
B) $ \frac{8}{89} $
C) $ \frac{49}{543} $
D) None of these
Show Answer
Answer:
Correct Answer: C
Solution:
Consider the function $ f(x)=\frac{x^{2}}{(x^{3}+200)} $ …..(i) $ f’(x)=x\frac{(400-x^{3})}{{{(x^{3}+200)}^{2}}}=0 $ When $ x={{(400)}^{1/3}}\ ,\ (\because x\ne 0) $
$ x={{(400)}^{1/3}}-h\Rightarrow f’(x)>0 $
$ x={{(400)}^{1/3}}+h\Rightarrow f’(x)<0 $
$ \therefore $ $ f(x) $ has maxima at $ x={{(400)}^{1/3}} $
Since $ 7<{{(400)}^{1/3}}<8, $ either $ a_7 $ or $ a_8 $ is the greatest term of the sequence.
$ \because a_7=\frac{49}{543} $ and $ a_8=\frac{8}{89} $ and $ \frac{49}{543}>\frac{8}{89} $
$ \therefore $ $ a_7=\frac{49}{543} $ is the greatest term.
BETA
