Let $P(n): 1^2+2^2+3^2+\ldots+n^2 = \frac{6(n-1)(n-2) \ldots(n-2020)+2n^3+3n^2+n}{6}$,for all $n \in N$. Then which of the following is correct?

Prove that $1^{2} + 2^{2} + \ldots + n^{2} > \frac{n^{3}}{3}$ for all $n \in N$.

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

Using mathematical induction,prove that $\frac{d}{dx}(x^n) = nx^{n-1}$ for all positive integers $n$.

Prove the following by using the principle of mathematical induction for all $n \in N:$
$4+8+12+\ldots+4n = 2n(n+1)$

Use the Principle of Mathematical Induction to prove that for all $n \in N$:
$\cos \alpha + \cos (\alpha + \beta) + \cos (\alpha + 2\beta) + \ldots + \cos [\alpha + (n-1)\beta] = \frac{\cos \left[\alpha + \left(\frac{n-1}{2}\right) \beta\right] \sin \left(\frac{n\beta}{2}\right)}{\sin \left(\frac{\beta}{2}\right)}$