For every positive integer $n$,${2^n} < n!$ when

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

Given ${U_{n + 1}} = 3{U_n} - 2{U_{n - 1}}$ and ${U_0} = 2$,${U_1} = 3$,the value of ${U_n}$ for all $n \in N$ is

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 $n \geq 2$.

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

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