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

Use the Principle of Mathematical Induction to prove that $\frac{1}{n+1} + \frac{1}{n+2} + \ldots + \frac{1}{2n} > \frac{13}{24}$ for all natural numbers $n > 1$.

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

Prove that the following inequality holds for all $n \in N$ by using the principle of mathematical induction:
$(2n + 7) < (n + 3)^{2}$

Prove that for all $n \in N$,$3^{2n+2} - 8n - 9$ is divisible by $8$ using the principle of mathematical induction.

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 $n$.