For every natural number $n$,which of the following inequalities is true?

Consider the statement $P(n): n^2 - n + 37$ is prime. Which one of the following is true?

Prove that for all $n \in N$,$41^{n}-14^{n}$ is a multiple of $27$ using the principle of mathematical induction.

Prove the statement by the Principle of Mathematical Induction:
$2n < (n+2)!$ for all natural numbers $n$.

For every positive integer $n$,${2^n} < n!$ when

Prove the following by using the principle of mathematical induction for all $n \in N$ :
$1^{3}+2^{3}+3^{3}+\ldots+n^{3}=\left(\frac{n(n+1)}{2}\right)^{2}$