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

Prove that $2 \cdot 7^{n} + 3 \cdot 5^{n} - 5$ is divisible by $24$ for all $n \in N$.

Prove the following by using the principle of mathematical induction for all $n \in N$:
$1+3+3^{2}+\ldots+3^{n-1}=\frac{3^{n}-1}{2}$

Use the Principle of Mathematical Induction to prove that $\cos \theta \cos 2 \theta \cos 2^{2} \theta \ldots \cos 2^{n-1} \theta = \frac{\sin 2^{n} \theta}{2^{n} \sin \theta}$ for all $n \in N$.

Using mathematical induction,the numbers $a_n$ are defined by $a_0=1$ and $a_{n+1}=3n^2+n+a_n$ for $n \geq 0$. Then $a_n$ is equal to:

Prove the statement by the Principle of Mathematical Induction: $n^{2} < 2^{n}$ for all natural numbers $n \geq 5$.