A student eats every day at the university restaurant. This restaurant offers vegetarian and non-vegetarian dishes.

\begin{itemize}
  \item When on a given day the student has chosen a vegetarian dish, the probability that he chooses a vegetarian dish the next day is 0.9.
  \item When on a given day the student has chosen a non-vegetarian dish, the probability that he chooses a vegetarian dish the next day is 0.7.
\end{itemize}

For any natural number $n$, we denote by $V _ { n }$ the event ``the student chose a vegetarian dish on the $n ^ { \mathrm { th } }$ day'' and $p _ { n }$ the probability of $V _ { n }$.\\
On the first day of the semester, the student chose the vegetarian dish. Thus $p _ { 1 } = 1$.

\begin{enumerate}
  \item a. Indicate the value of $p _ { 2 }$.\\
b. Show that $p _ { 3 } = 0.88$. You may use a probability tree.\\
c. Given that on the 3rd day the student chose a vegetarian dish, what is the probability that he chose a non-vegetarian dish the previous day?\\
Round the result to $10 ^ { - 2 }$.
  \item Copy and complete the probability tree.
  \item Justify that, for any natural number $n \geqslant 1 , p _ { n + 1 } = 0.2 p _ { n } + 0.7$.
  \item We wish to have the list of the first terms of the sequence $( p _ { n } )$ for $n \geqslant 1$. For this, we use a function called meals programmed in Python language, of which three versions are proposed below.
\end{enumerate}

\begin{verbatim}
    Program 1
def meals(n):
    p=1
    L= [p]
    for k in range(1,n):
        p = 0.2*p+0.7
        L. append(p)
        return(L)
\end{verbatim}

\begin{verbatim}
    Program 2
def meals(n):
    p=1
    L= [p]
    for k in range(1,n+1):
        p = 0.2*p+0.7
        L. append(p)
    return(L)
\end{verbatim}

\begin{verbatim}
    Program 3
def meals(n):
    p=1
    L=[p]
    for k in range(1,n):
        p = 0.2*p+0.7
        L.append(p+1)
    return(L)
\end{verbatim}

a. Which of these programs allows displaying the first $n$ terms of the sequence $\left( p _ { n } \right)$? No justification is required.\\
b. With the program chosen in question a., give the result displayed for $n = 5$.\\
4. Prove by induction that, for any natural number $n \geqslant 1 , p _ { n } = 0.125 \times 0.2 ^ { n - 1 } + 0.875$.\\
5. Deduce the limit of the sequence $\left( p _ { n } \right)$.