\textbf{(For candidates who have not followed the specialization course)}

We consider two sequences of real numbers $( d _ { n } )$ and $( a _ { n } )$ defined by $d _ { 0 } = 300$, $a _ { 0 } = 450$ and, for every natural number $n \geqslant 0$

$$\left\{ \begin{array} { l } 
d _ { n + 1 } = \frac { 1 } { 2 } d _ { n } + 100 \\
a _ { n + 1 } = \frac { 1 } { 2 } d _ { n } + \frac { 1 } { 2 } a _ { n } + 70
\end{array} \right.$$

\begin{enumerate}
  \item Calculate $d _ { 1 }$ and $a _ { 1 }$.
  \item It is desired to write an algorithm that allows displaying as output the values of $d _ { n }$ and $a _ { n }$ for an integer value of $n$ entered by the user. The following algorithm is proposed:

\begin{center}
\begin{tabular}{|l|l|}
\hline
Variables: & \begin{tabular}{l}
$n$ and $k$ are natural numbers \\
$D$ and $A$ are real numbers \\
\end{tabular} \\
\hline
Initialization: & \begin{tabular}{l}
$D$ takes the value 300 \\
$A$ takes the value 450 \\
Enter the value of $n$ \\
\end{tabular} \\
\hline
Processing: & \begin{tabular}{l}
For $k$ varying from 1 to $n$ \\
$D$ takes the value $\frac { D } { 2 } + 100$ \\
$A$ takes the value $\frac { A } { 2 } + \frac { D } { 2 } + 70$ \\
End for \\
\end{tabular} \\
\hline
Output: & \begin{tabular}{l}
Display $D$ \\
Display $A$ \\
\end{tabular} \\
\hline
\end{tabular}
\end{center}

a. What numbers are obtained as output of the algorithm for $n = 1$? Are these results consistent with those obtained in question 1?\\
b. Explain how to correct this algorithm so that it displays the desired results.
  \item a. For every natural number $n$, we set $e _ { n } = d _ { n } - 200$. Show that the sequence $( e _ { n } )$ is geometric.\\
b. Deduce the expression of $d _ { n }$ as a function of $n$.\\
c. Is the sequence $( d _ { n } )$ convergent? Justify.
  \item We admit that for every natural number $n$,

$$a _ { n } = 100 n \left( \frac { 1 } { 2 } \right) ^ { n } + 110 \left( \frac { 1 } { 2 } \right) ^ { n } + 340 .$$

a. Show that for every integer $n$ greater than or equal to 3, we have $2 n ^ { 2 } \geqslant ( n + 1 ) ^ { 2 }$.\\
b. Show by induction that for every integer $n$ greater than or equal to 4, $2 ^ { n } \geqslant n ^ { 2 }$.\\
c. Deduce that for every integer $n$ greater than or equal to 4,

$$0 \leqslant 100 n \left( \frac { 1 } { 2 } \right) ^ { n } \leqslant \frac { 100 } { n } .$$

d. Study the convergence of the sequence $\left( a _ { n } \right)$.
\end{enumerate}