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:

Prove that for all $n \in N$,$41^{n}-14^{n}$ is a multiple of $27$ using the principle of mathematical induction.

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 $n \in N$.

Prove that the following is true for all $n \in N$ using the principle of mathematical induction:
$10^{2n-1} + 1$ is divisible by $11$.

Let $P(n) = 3^{2n+1} + 2^{n+2}$ where $n \in N$. Then

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