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

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

Prove the following by using the principle of mathematical induction for all $n \in N$ where $n \geq 2$:
$\left(1-\frac{1}{2^{2}}\right)\left(1-\frac{1}{3^{2}}\right)\left(1-\frac{1}{4^{2}}\right) \ldots \left(1-\frac{1}{n^{2}}\right)=\frac{n+1}{2n}$

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

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

For every positive integral value of $n$,${3^n} > {n^3}$ when