Let $(u_n)$ be the sequence defined by its first term $u_0$ and, for every natural number $n$, by the relation $$u_{n+1} = a u_n + b \quad (a \text{ and } b \text{ non-zero real numbers such that } a \neq 1).$$ We set, for every natural number $n$, $\quad v_n = u_n - \dfrac{b}{1-a}$.
Prove that the sequence $(v_n)$ is geometric with common ratio $a$.
Deduce that if $a$ belongs to the interval $]-1\,;\,1[$, then the sequence $(u_n)$ has limit $\dfrac{b}{1-a}$.
Part B
In March 2015, Max buys a green plant measuring 80 cm. He is advised to prune it every year, in March, by cutting a quarter of its height. The plant will then grow 30 cm over the following twelve months. As soon as he gets home, Max prunes his plant.
What will be the height of the plant in March 2016 before Max prunes it?
For every natural number $n$, we denote by $h_n$ the height of the plant, before pruning, in March of the year $(2015 + n)$. a. Justify that, for every natural number $n$, $\quad h_{n+1} = 0.75\,h_n + 30$. b. Conjecture using a calculator the direction of variation of the sequence $(h_n)$. Prove this conjecture (you may use a proof by induction). c. Is the sequence $(h_n)$ convergent? Justify your answer.
\section*{Exercise 2 (5 points) -- Common to all candidates}
\section*{Part A}
Let $(u_n)$ be the sequence defined by its first term $u_0$ and, for every natural number $n$, by the relation
$$u_{n+1} = a u_n + b \quad (a \text{ and } b \text{ non-zero real numbers such that } a \neq 1).$$
We set, for every natural number $n$, $\quad v_n = u_n - \dfrac{b}{1-a}$.
\begin{enumerate}
\item Prove that the sequence $(v_n)$ is geometric with common ratio $a$.
\item Deduce that if $a$ belongs to the interval $]-1\,;\,1[$, then the sequence $(u_n)$ has limit $\dfrac{b}{1-a}$.
\end{enumerate}
\section*{Part B}
In March 2015, Max buys a green plant measuring 80 cm. He is advised to prune it every year, in March, by cutting a quarter of its height. The plant will then grow 30 cm over the following twelve months. As soon as he gets home, Max prunes his plant.
\begin{enumerate}
\item What will be the height of the plant in March 2016 before Max prunes it?
\item For every natural number $n$, we denote by $h_n$ the height of the plant, before pruning, in March of the year $(2015 + n)$.\\
a. Justify that, for every natural number $n$, $\quad h_{n+1} = 0.75\,h_n + 30$.\\
b. Conjecture using a calculator the direction of variation of the sequence $(h_n)$. Prove this conjecture (you may use a proof by induction).\\
c. Is the sequence $(h_n)$ convergent? Justify your answer.
\end{enumerate}