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

Let $P(n) = 3^{2n+1} + 2^{n+2}$ where $n \in N$. Then

Prove the statement by the Principle of Mathematical Induction: $2+4+6+\ldots+2n = n^2+n$ for all natural numbers $n$.

Use the Principle of Mathematical Induction to show that for a sequence $b_{0}, b_{1}, b_{2}, \ldots$ defined by $b_{0}=5$ and $b_{k}=4+b_{k-1}$ for all natural numbers $k$,the general term is $b_{n}=5+4n$ for all natural numbers $n$.

Prove the statement by the Principle of Mathematical Induction: $n^{3}-n$ is divisible by $6$ for each natural number $n \geq 2$.

Let $S(k) = 1 + 3 + 5 + \dots + (2k - 1) = 3 + k^2$. Then which of the following is true?