In a country with a constant population equal to 120 million, the inhabitants live either in rural areas or in cities. Population movements can be modeled as follows:
\begin{itemize}
  \item in 2010, the population consists of 90 million rural inhabitants and 30 million city dwellers;
  \item each year, $10\%$ of rural inhabitants migrate to the city;
  \item each year, $5\%$ of city dwellers migrate to rural areas.
\end{itemize}
For any natural integer $n$, we denote:
\begin{itemize}
  \item $u_n$ the population in rural areas, in the year $2010 + n$, expressed in millions of inhabitants;
  \item $v_n$ the population in cities, in the year $2010 + n$, expressed in millions of inhabitants.
\end{itemize}
We have $u_0 = 90$ and $v_0 = 30$.

\textbf{Part A}
\begin{enumerate}
  \item Translate the fact that the total population is constant by a relation linking $u_n$ and $v_n$.
  \item What formulas can be entered in cells B3 and C3 which, copied downward, allow us to obtain the spreadsheet showing the evolution of $(u_n)$ and $(v_n)$?
  \item What conjectures can be made concerning the long-term evolution of this population?
\end{enumerate}

\textbf{Part B}

Deduce from the recurrence relations that, for every natural integer $n$, $R_n = 50 \times 0.85^n + 40$ and determine the expression of $C_n$ as a function of $n$.\\
b. Determine the limit of $R_n$ and of $C_n$ when $n$ tends towards $+\infty$. What can we conclude from this for the population studied?\\
6. a. Complete the algorithm so that it displays the number of years after which the urban population will exceed the rural population.\\
b. By solving the inequality with unknown $n$,
$$50 \times 0.85^n + 40 < 80 - 50 \times 0.85^n,$$
find again the value displayed by the algorithm.