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 $n \in N:$
$3 \times 6 + 6 \times 9 + 9 \times 12 + \ldots + (3n)(3n + 3) = 3n(n + 1)(n + 2)$

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

Let $P(n) : 3^n < n!$ for $n \in N$ be true for $n \geq \lambda$. Find the smallest value of $\lambda$.

For all $n \ge 1,$ prove that $1^{2}+2^{2}+3^{2}+4^{2}+\ldots+n^{2}=\frac{n(n+1)(2 n+1)}{6}$

Use the Principle of Mathematical Induction to prove that for all $n \in N$:
$\sin \theta + \sin 2\theta + \ldots + \sin n\theta = \frac{\sin \frac{n\theta}{2} \sin \frac{(n+1)\theta}{2}}{\sin \frac{\theta}{2}}$