For what natural numbers $n\in N$, the inequality $2^n>n+1$ is valid?

Given that, $2^n>n+1$

For $n=1$, $2^1>1+1\Rightarrow 2>2$ is not true

For $n=2$, $2^2>2+1\Rightarrow 4>3$ is True.

So, Given inequality is true for $n=2$.

Assume that  $2^m>m+1$ is true for $m>2$.

But $2^m>1$ for all $m>1$

$\Rightarrow 2^m+2^m>m+1+1$

$\Rightarrow 2^{m+1}>\left(m+1\right)+1$

Hence, the inequality holds for $n=m+1$

It follows by mathematical induction that the given inequality holds for all natural numbers $n\ge 2$.