Question

The least value of $n,$ for which $n!<{\left[\frac{n+1}{2}\right]}^n$ is true for odd natural values of $n$ , is
(where $[.]$ denotes the greatest integer function
(use principle of mathematical induction)

• A. $1$
• B. $5$
• C. $3$
• D. $7$

Answer 1

The correct option is C $3$

As, $P\left(n\right):n!<{\left[\frac{n+1}{2}\right]}^n$ is true for odd natural values of $n.$
On verifying for $n=1,3,\cdots$
For n = 1, $1!<{\left[\frac{1+1}{2}\right]}^1$
i.e. $1!<1$ is not true.

For n = 3, $3!<{\left[\frac{3+1}{2}\right]}^3$
i.e. $3!<8,$ is true
So, minimum value of $n=3$