For any $n \in N$,$4^n+15n-1$ is divisible by

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

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

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

Prove that for all $n \in N$,$41^{n}-14^{n}$ is a multiple of $27$ using the principle of mathematical induction.

Prove the following by using the principle of mathematical induction for all $n \in N:$
$4+8+12+\ldots+4n = 2n(n+1)$