Question

The smallest 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 $2$

Smallest positive integers includes $1,2,3,...\infty$
Substituting $n=1$ in the above inequality, we get
$1!<{\left[\frac{2}{2}\right]}^1$
$1<\mathrm{1...}$

The inequality does not holds.
Substituting $n=2$ in the above inequality.
$2!<{\left[\frac{3}{2}\right]}^2$
$2<(1.5{)}^2$
$2<2.25$
The inequality holds for $n=2.$