For all natural numbers $n$,$3(5^{2n+1}) + 2^{3n+1}$ is divisible by:

Prove the statement by the Principle of Mathematical Induction: $n(n^{2}+5)$ is divisible by $6$,for each natural number $n$.

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

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 $n$.

If $n \in N$,then the statement $8n + 16 \leq 2^n$ is true for:

Let $P(n) : 3^n < n!$ for $n \in N$ be true for $n \geq \lambda$. Find the smallest value of $\lambda$.