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

Prove the statement by the Principle of Mathematical Induction: $4^{n}-1$ is divisible by $3$,for each natural number $n$.

Given ${U_{n + 1}} = 3{U_n} - 2{U_{n - 1}}$ and ${U_0} = 2$,${U_1} = 3$,the value of ${U_n}$ for all $n \in N$ is

Let $P(n): 1^2+2^2+3^2+\ldots+n^2 = \frac{6(n-1)(n-2) \ldots(n-2020)+2n^3+3n^2+n}{6}$,for all $n \in N$. Then which of the following is correct?

Prove the following by using the principle of mathematical induction for all $n \in N:$
$\frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + \ldots + \frac{1}{(2n-1)(2n+1)} = \frac{n}{2n+1}$

For all $n \ge 1,$ prove that $1^{2}+2^{2}+3^{2}+4^{2}+\ldots+n^{2}=\frac{n(n+1)(2 n+1)}{6}$