Mathematical Reasoning

Given that $n\left(n^2+3\right)$ is divisible by $k$, for all $n \in N$

For $n=1,1\left(1^2+3\right)=1.4=4$ is true, when $k$ is 2 and 4 .

Let assume for $n=2,2\left(2^2+3\right)=14$ is true for 2 but not 4 .

$\therefore$ Maximum value of $k$ is 2 .

Clearly, $2^n > n + 1$ is valid for, $\forall n\ge 2$.

$\begin{array}{ll} \text { } 8 n+16 \leq 2^n, n \in N \\ \Rightarrow 8(n+2) \leq 2^n \\ \Rightarrow n+2 \leq 2^{n-3} \end{array}$

Clearly, for $n=6$

$6+2 \leq 2^{6-3}$ or $8 \leq 8$ is true.