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

Prove the statement by the Principle of Mathematical Induction: $1+2+2^{2}+\ldots+2^{n}=2^{n+1}-1$ for all natural numbers $n$.

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

Let $P(n): 2+2^2+2^3+\ldots+2^n=2^{n+1}-2, n \in N$. Then,

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

Using mathematical induction,prove that $\frac{d}{dx}(x^n) = nx^{n-1}$ for all positive integers $n$.