We consider the sequence $(u_n)$ defined by: $$\left\{\begin{aligned} u_1 &= \frac{1}{\mathrm{e}} \\ u_{n+1} &= \frac{1}{\mathrm{e}}\left(1 + \frac{1}{n}\right)u_n \text{ for all integer } n \geqslant 1. \end{aligned}\right.$$

1. Calculate the exact values of $u_2$ and $u_3$. Details of calculations should be shown.
2. We consider a function written in Python language which, for a given natural integer $n$, displays the term $u_n$. Complete lines $L_2$ and $L_4$ of this program. \begin{verbatim} L1 def u(n): L2 ................. L3 for i in range(1, n): L4 u=................. L5 return u \end{verbatim}
3. We admit that all terms of the sequence $(u_n)$ are strictly positive. a. Show that for all non-zero natural integer $n$, we have: $1 + \dfrac{1}{n} \leqslant \mathrm{e}$. b. Deduce that the sequence $(u_n)$ is decreasing. c. Is the sequence $(u_n)$ convergent? Justify your answer.
4. a. Show by induction that for all non-zero natural integer, we have: $u_n = \dfrac{n}{\mathrm{e}^n}$. b. Deduce, if it exists, the limit of the sequence $(u_n)$.

The sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } = 3 , a _ { n + 1 } = 3 a _ { n } - 4 n$ .
(1) Calculate $a _ { 2 } , a _ { 3 }$ , conjecture the general term formula of $\left\{ a _ { n } \right\}$ and prove it;
(2) Find the sum $S _ { n }$ of the first $n$ terms of the sequence $\left\{ 2 ^ { n } a _ { n } \right\}$ .