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

Using mathematical induction,the numbers $a_n$ are defined by:
$a_0 = 1, a_{n+1} = 3n^2 + n + a_n, (n \geq 0)$.
Then,$a_n$ is equal to:

If $P(n): 2^{n} < n!$,then the smallest positive integer for which $P(n)$ is true,is

Prove the statement by the Principle of Mathematical Induction: For any natural number $n$,$7^{n}-2^{n}$ is divisible by $5$.

Use the Principle of Mathematical Induction to show that for a sequence $d_{1}, d_{2}, d_{3}, \ldots$ defined by $d_{1}=2$ and $d_{k}=\frac{d_{k-1}}{k}$ for all $k \geq 2$,the general term is $d_{n}=\frac{2}{n!}$ for all $n \in N$.

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