Newton's law of cooling states that the rate of change of the temperature of a body is proportional to the difference between the temperature of this body and that of the surrounding environment.

A cup of coffee is served at an initial temperature of $80^{\circ}\mathrm{C}$ in an environment whose temperature, expressed in degrees Celsius, assumed to be constant, is denoted $M$.

In this part, for any natural number $n$, we denote $T_n$ the temperature of the coffee at instant $n$, with $T_n$ expressed in degrees Celsius and $n$ in minutes. Thus $T_0 = 80$.

We model Newton's law between any two consecutive minutes $n$ and $n+1$ by the equality:
$$T_{n+1} - T_n = k(T_n - M)$$
where $k$ is a real constant.

In the rest of part A, we choose $M = 10$ and $k = -0{,}2$. Thus, for any natural number $n$, we have: $T_{n+1} - T_n = -0{,}2(T_n - 10)$.

\begin{enumerate}
  \item Based on the context, can we conjecture the direction of variation of the sequence $(T_n)$?
  \item Show that for any natural number $n$: $T_{n+1} = 0{,}8T_n + 2$.
  \item We set, for any natural number $n$: $u_n = T_n - 10$.\\
  a. Show that $(u_n)$ is a geometric sequence. Specify its common ratio and its first term $u_0$.\\
  b. Show that, for any natural number $n$, we have: $T_n = 70 \times 0{,}8^n + 10$.\\
  c. Determine the limit of the sequence $(T_n)$.
  \item Consider the following algorithm:
\begin{verbatim}
While $T \geqslant 40$
    $T \leftarrow 0,8T + 2$
    $n \leftarrow n + 1$
End While
\end{verbatim}
  a. Initially, we assign the value 80 to the variable $T$ and the value 0 to the variable $n$. What numerical value does the variable $n$ contain at the end of the algorithm's execution?\\
  b. Interpret this value in the context of the exercise.
\end{enumerate}