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

If $n$ is a positive integer,then $n^{3}+2n$ is divisible by

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 $n \in N:$
$7^{n}-3^{n}$ is divisible by $4$.

If $P(n) = 2 + 4 + 6 + \dots + 2n$,$n \in N$,then $P(k) = k(k + 1) + 2 \implies P(k + 1) = (k + 1)(k + 2) + 2$ for all $k \in N$. So we can conclude that $P(k) = k(k + 1) + 2$ for all $k \in N$ is true. What can we conclude about $P(n) = n(n + 1) + 2$ for all $n \in N$?

Let $P(n)$ be a statement and let $P(n) \implies P(n + 1)$ for all natural numbers $n$. Then $P(n)$ is true for: