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

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

Use the Principle of Mathematical Induction to show that $\frac{n^{5}}{5}+\frac{n^{3}}{3}+\frac{7n}{15}$ is a natural number for all $n \in N$.

Prove the following by using the principle of mathematical induction for all $n \in N:$
$3^{2n} - 1$ is divisible by $8$.

Use the Principle of Mathematical Induction to prove that the number of subsets of a set containing $n$ distinct elements is $2^{n}$,for all $n \in N$.

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