Question

If $p\left(n\right):n^2<2^n$ is true for $n\in N-\left\{1\right\}.$ then the minimum value of $n=$
(use principle of mathematical induction)

Answer 1

Given :
$p\left(n\right):n^2<2^n$ is true for $n\in N-\left\{1\right\}$
So, $P\left(n\right)$ is true for $n=2,3,\cdots$
For $n=2,P\left(2\right):4<4$ (false)
For $n=3,P\left(3\right):9<8$ (false)
For $n=4,P\left(4\right):16<16$ (false)
For $n=5,P\left(5\right):25<32$ (true)
So, minimum value of $n=5$

$p\left(n\right):n^2<2^n;n\ge 5$ is true for $n=k$
For $k=n+1$
$p(n+1):(n+1{)}^2<2^{n+1};n\ge 5$
$p(n+1):n^2+2n+1<2\cdot 2^n;n\ge 5$
and $2n+1<2^n;n\ge 5$
$\therefore$ True for $k=n+1$