Question

The least positive integer $n$, for which $n!<{\left(\frac{n+1}{2}\right)}^n$ holds, is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is B $2$

For $n=1$, $1!<{\left(1\right)}^1$

that is, $1<1$ is not true.

For $n=2$, $2!<{\left(\frac{3}{2}\right)}^2$

that is, $2<\frac{9}{4}$ is true.

By induction, $n!<{\left(\frac{n+1}{2}\right)}^n\forall n\in N$ for $n\ge 2$.

Therefore, least positive integer is $n=2$.