For each $n \in N$,which of the following statements is correct?

Prove the following by using the principle of mathematical induction for all $n \in N$:
$\left(1+\frac{3}{1}\right)\left(1+\frac{5}{4}\right)\left(1+\frac{7}{9}\right) \dots \left(1+\frac{2n+1}{n^{2}}\right)=(n+1)^{2}$

Prove the statement by the Principle of Mathematical Induction: $n^{3}-n$ is divisible by $6$ for each natural number $n \geq 2$.

Prove the statement by the Principle of Mathematical Induction: For any natural number $n$,$x^{n}-y^{n}$ is divisible by $x-y$,where $x$ and $y$ are any integers with $x \neq y$.

Use the Principle of Mathematical Induction to show that for a sequence $b_{0}, b_{1}, b_{2}, \ldots$ defined by $b_{0}=5$ and $b_{k}=4+b_{k-1}$ for all natural numbers $k$,the general term is $b_{n}=5+4n$ for all natural numbers $n$.

Prove that $1^{2} + 2^{2} + \ldots + n^{2} > \frac{n^{3}}{3}$ for all $n \in N$.