\textbf{Part B - Second model}

After studying a larger collection of data over the last 50 years, another modelling appears more relevant:
\begin{itemize}
  \item if the El Niño phenomenon is dominant in one year, then the probability that it is still dominant the following year is 0.5
  \item on the other hand, if the El Niño phenomenon is not dominant in one year, then the probability that it is dominant the following year is 0.3.
\end{itemize}

We consider that the reference year is 2023.\\
We denote for every natural integer $n$:
\begin{itemize}
  \item $E _ { n }$ the event ``the El Niño phenomenon is dominant in the year $2023 + n$ '';
  \item $p _ { n }$ the probability of the event $E _ { n }$.
\end{itemize}

In 2023, El Niño was not dominant. We thus have $p _ { 0 } = 0$.

\begin{enumerate}
  \item Let $n$ be a natural integer. Copy and complete the following weighted tree.
  \item Justify that $p _ { 1 } = 0.3$.
  \item Using the tree, show that, for every natural integer $n$, we have:
$$p _ { n + 1 } = 0.2 p _ { n } + 0.3$$
  \item a. Conjecture the variations and the possible limit of the sequence $( p _ { n } )$.\\
b. Show by induction that, for every natural integer $n$, we have: $p _ { n } \leqslant \frac { 3 } { 8 }$.\\
c. Determine the direction of variation of the sequence $\left( p _ { n } \right)$.\\
d. Deduce the convergence of the sequence $\left( p _ { n } \right)$.
  \item Let $\left( u _ { n } \right)$ be the sequence defined by $u _ { n } = p _ { n } - \frac { 3 } { 8 }$ for every natural integer $n$.\\
a. Show that the sequence $( u _ { n } )$ is geometric with ratio 0.2 and specify its first term.\\
b. Show that, for every natural integer $n$, we have:
$$p _ { n } = \frac { 3 } { 8 } \left( 1 - 0.2 ^ { n } \right) .$$
c. Calculate the limit of the sequence $\left( p _ { n } \right)$.\\
d. Interpret this result in the context of the exercise.
\end{enumerate}