This exercise contains 5 statements. For each statement, answer TRUE or FALSE by justifying the answer. Any lack of justification or incorrect justification will not be taken into account in the grading.

\textbf{Part 1}

We consider the sequence $(u_n)$ defined by:
$$u_0 = 10 \text{ and for all natural integer } n,\ u_{n+1} = \frac{1}{3}u_n + 2.$$

\begin{enumerate}
  \item Statement 1: The sequence $(u_n)$ is decreasing and bounded below by 0.
  \item Statement 2: $\lim_{n \rightarrow +\infty} u_n = 0$.
  \item Statement 3: The sequence $(v_n)$ defined for all natural integer $n$ by $v_n = u_n - 3$ is geometric.
\end{enumerate}

\textbf{Part 2}

We consider the differential equation $(E): y' = \frac{3}{2}y + 2$ with unknown $y$, a function defined and differentiable on $\mathbb{R}$.

\begin{enumerate}
  \item Statement 4: There exists a constant function that is a solution to the differential equation $(E)$.
  \item In an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$ we denote $\mathscr{C}_f$ the representative curve of the function $f$ solution of $(E)$ such that $f(0) = 0$.\\
  Statement 5: The tangent line at the point with abscissa 1 of $\mathscr{C}_f$ has slope $2\mathrm{e}^{\frac{3}{2}}$.
\end{enumerate}