If $n$ is a positive integer,then $2 \cdot 4^{2n+1} + 3^{3n+1}$ is divisible by

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?

If $n \in N$,then the statement $8n + 16 \leq 2^n$ is true for:

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 $n \geq 2$.

Consider the statement: $P(n): n^2 - n + 41$ is prime. Then which one of the following is true?

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