If $P(n): 2^{n} < n!$,then the smallest positive integer for which $P(n)$ is true,is

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

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

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

Given ${U_{n + 1}} = 3{U_n} - 2{U_{n - 1}}$ and ${U_0} = 2$,${U_1} = 3$,the value of ${U_n}$ for all $n \in N$ is

Let $P(n) = 3^{2n+1} + 2^{n+2}$ where $n \in N$. Then