A) Less than \[{{\left( \frac{n+1}{2} \right)}^{3}}\]
B) Greater than \[{{\left( \frac{n+1}{2} \right)}^{3}}\]
C) Less than \[{{(n!)}^{3}}\]
D) Greater than \[{{(n!)}^{3}}\,\]
Correct Answer: D
Solution :
\[y={{n}^{n}}{{\left( \frac{n+1}{2} \right)}^{2n}}\] Put n = 2, \[y={{2}^{2}}{{\left( \frac{3}{2} \right)}^{4}}=4\,.\,\frac{81}{8\times 2}=\frac{81}{4}\tilde{-}\]20 Option (a) \[={{\left( \frac{n+1}{2} \right)}^{3}}=\frac{27}{8}<y\] Option (b) \[={{\left( \frac{n+1}{2} \right)}^{3}}=\frac{27}{8}<y\] Option (c) \[={{(2!)}^{3}}=8<y\] Option (d)\[={{(2!)}^{3}}=8<y\].
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