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

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

Prove the following by using the principle of mathematical induction for all $n \in N$:
$\frac{1}{2 \times 5} + \frac{1}{5 \times 8} + \frac{1}{8 \times 11} + \ldots + \frac{1}{(3n-1)(3n+2)} = \frac{n}{6n+4}$

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

Consider the statement $P(n): n^2 - n + 37$ is prime. Which one of the following is true?

Use the Principle of Mathematical Induction to show that for a sequence $a_{1}, a_{2}, a_{3}, \ldots$ defined by $a_{1}=3$ and $a_{k}=7 a_{k-1}$ for all natural numbers $k > 1$,the general term is $a_{n}=3 \cdot 7^{n-1}$ for all $n \in N$.