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

The values of the natural numbers $n$ for which the inequality $2^n > 2n + 1$ is valid are:

Prove the following by using the principle of mathematical induction for all $n \in N$:
$1 \cdot 3 + 3 \cdot 5 + 5 \cdot 7 + \ldots + (2n - 1)(2n + 1) = \frac{n(4n^2 + 6n - 1)}{3}$

Prove the following by using the principle of mathematical induction for all $n \in N$:
$\frac{1}{1 \cdot 4} + \frac{1}{4 \cdot 7} + \frac{1}{7 \cdot 10} + \ldots + \frac{1}{(3n-2)(3n+1)} = \frac{n}{3n+1}$

Prove the statement by the Principle of Mathematical Induction: $3^{2n} - 1$ is divisible by $8$ for all natural numbers $n$.

For any $n \in N$,$4^n+15n-1$ is divisible by