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 prove that for all $n \in N$:
$\cos \alpha + \cos (\alpha + \beta) + \cos (\alpha + 2\beta) + \ldots + \cos [\alpha + (n-1)\beta] = \frac{\cos \left[\alpha + \left(\frac{n-1}{2}\right) \beta\right] \sin \left(\frac{n\beta}{2}\right)}{\sin \left(\frac{\beta}{2}\right)}$

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

Prove the statement by the Principle of Mathematical Induction :
$1+5+9+\ldots+(4 n-3)=n(2 n-1)$ for all natural numbers $n$.

Let $P(n)$ denote the statement that $n^2 + n$ is odd. It is seen that $P(n) \Rightarrow P(n + 1)$. $P(n)$ is true for all:

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