Let $P(n): 2+2^2+2^3+\ldots+2^n=2^{n+1}-2, n \in N$. Then,

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 $n \geq 2$.

Prove the following by using the principle of mathematical induction for all $n \in N$:
$\frac{1}{2 \times 5} + \frac{1}{5 \times 8} + \frac{1}{8 \times 11} + \ldots + \frac{1}{(3n-1)(3n+2)} = \frac{n}{6n+4}$

For each $n \in N$,which of the following statements is correct?

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

Prove the statement by the Principle of Mathematical Induction:
$2n < (n+2)!$ for all natural numbers $n$.