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

For all $n \in \mathbb{N}$,if $1^2+2^2+3^2+\ldots+n^2 > x$,then $x=$

When $P$ is a natural number,then ${P^{n + 1}} + {(P + 1)^{2n - 1}}$ is divisible by

Prove the statement by the Principle of Mathematical Induction: $n^{3}-7n+3$ is divisible by $3$ for all natural numbers $n$.

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

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