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 the statement by the Principle of Mathematical Induction :
$1+5+9+\ldots+(4 n-3)=n(2 n-1)$ for all natural numbers $n$.

Prove the statement by the Principle of Mathematical Induction: $1+2+2^{2}+\ldots+2^{n}=2^{n+1}-1$ for all natural numbers $n$.

Prove the statement by the Principle of Mathematical Induction: $2+4+6+\ldots+2n = n^2+n$ for all natural numbers $n$.

Prove the following by using the principle of mathematical induction for all $n \in N$:
$\frac{1}{1 \cdot 4} + \frac{1}{4 \cdot 7} + \frac{1}{7 \cdot 10} + \ldots + \frac{1}{(3n-2)(3n+1)} = \frac{n}{3n+1}$

Prove the statement by the Principle of Mathematical Induction: $\sqrt{n} < \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \ldots + \frac{1}{\sqrt{n}}$ for all natural numbers $n \geq 2$.