For each $n \in N$,which of the following statements is correct?

For every positive integer $n,$ prove that $7^{n}-3^{n}$ is divisible by $4.$

Prove the statement by the Principle of Mathematical Induction: $2^{3n} - 1$ is divisible by $7$ for all natural numbers $n$.

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

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

Prove the following by using the principle of mathematical induction for all $n \in N$:
$\frac{1}{1 \cdot 4} + \frac{1}{4 \cdot 7} + \frac{1}{7 \cdot 10} + \ldots + \frac{1}{(3n-2)(3n+1)} = \frac{n}{3n+1}$