\section*{Exercise 4 (5 points)}
\section*{Candidates who have not followed the specialization course}

A radio-controlled scooter moves in a straight line at the constant speed of $1\,\mathrm{m.s}^{-1}$. It is pursued by a dog that moves at the same speed.\\
We represent the situation from above in an orthonormal coordinate system of the plane with unit 1 meter. The origin of this coordinate system is the initial position of the dog. The scooter is represented by a point belonging to the line with equation $x = 5$. It moves on this line in the direction of increasing ordinates.

\section*{Part A - Modeling using a sequence}
At the initial instant, the scooter is represented by the point $S_0$. The dog pursuing it is represented by the point $M_0$. We consider that at each second, the dog instantly orients itself in the direction of the scooter and moves in a straight line over a distance of 1 meter.\\
We then model the trajectories of the dog and the scooter by two sequences of points denoted $(M_n)$ and $(S_n)$. After $n$ seconds, the coordinates of point $S_n$ are $(5; n)$. We denote $(x_n; y_n)$ the coordinates of point $M_n$.

\begin{enumerate}
  \item Construct on graph $\mathrm{n}^\circ 1$ given in the appendix the points $M_2$ and $M_3$.
  \item We denote $d_n$ the distance between the dog and the scooter $n$ seconds after the start of the pursuit, $d_n = M_nS_n$.\\
Calculate $d_0$ and $d_1$.
  \item Justify that the point $M_2$ has coordinates $\left(1 + \frac{4}{\sqrt{17}}; \frac{1}{\sqrt{17}}\right)$.
  \item We admit that, for every natural integer $n$:
$$\left\{\begin{array}{l} x_{n+1} = x_n + \dfrac{5 - x_n}{d_n} \\[6pt] y_{n+1} = y_n + \dfrac{n - y_n}{d_n} \end{array}\right.$$
a. The table below, obtained using a spreadsheet, gives the coordinates of points $M_n$ and $S_n$ as well as the distance $d_n$ as a function of $n$. What formulas should be written in cells C5 and F5 and copied downward to fill columns C and F?
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
 & A & B & C & D & E & F \\
\hline
1 & $n$ & \multicolumn{2}{|c|}{$M_n$} & \multicolumn{2}{|c|}{$S_n$} & $d_n$ \\
\hline
2 &  & $x_n$ & $y_n$ & 5 & n &  \\
\hline
3 & 0 & 0 & 0 & 5 & 0 & 5 \\
\hline
4 & 1 & 1 & 0 & 5 & 1 & 4.12310563 \\
\hline
5 & 2 & 1.9701425 & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ \\
\hline
\end{tabular}
\end{center}
\end{enumerate}