A) \[n!>{{\left( \frac{n+1}{2} \right)}^{n}}\]
B) \[n!\ge {{\left( \frac{n+1}{2} \right)}^{n}}\]
C) \[n!<{{\left( \frac{n+1}{2} \right)}^{n}}\]
D) None of these
Correct Answer: C
Solution :
| [c] when n =2 then \[{{\left( \frac{n+1}{2} \right)}^{n}}=\frac{9}{4}\] |
| \[\Rightarrow n!<{{\left( \frac{n+1}{2} \right)}^{n}}\] |
| When \[n=3,\] then \[n!=6,{{\left( \frac{n+1}{2} \right)}^{n}}=8\] |
| \[\Rightarrow n!<{{\left( \frac{n+1}{2} \right)}^{n}}\] |
| When \[n=4,\] then \[n!=24.\] |
| \[{{\left( \frac{n+1}{2} \right)}^{n}}=\frac{625}{16}\Rightarrow n!<{{\left( \frac{n+1}{2} \right)}^{n}}\] |
| \[\therefore \] It is seen that \[n!<{{\left( \frac{n+1}{2} \right)}^{n}}\] |
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