For every natural number $n$,which of the following inequalities is true?

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 $n \in N$:
$1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + n(n+1) = \frac{n(n+1)(n+2)}{3}$

For what natural numbers $n \in N$,is the inequality $2^n > n+1$ valid?

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

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