Exercise 4 (5 points) -- Candidates who have NOT chosen the specialization option Part A Consider the sequence $(u_{n})$ defined by: $u_{0} = 2$ and, for all natural integers $n$: $$u_{n+1} = \frac{1 + 3u_{n}}{3 + u_{n}}$$ It is admitted that all terms of this sequence are defined and strictly positive.
Prove by induction that, for all natural integers $n$, we have: $u_{n} > 1$.
a. Establish that, for all natural integers $n$, we have: $u_{n+1} - u_{n} = \frac{(1 - u_{n})(1 + u_{n})}{3 + u_{n}}$. b. Determine the direction of variation of the sequence $(u_{n})$. Deduce that the sequence $(u_{n})$ converges.
Part B Consider the sequence $(u_{n})$ defined by: $u_{0} = 2$ and, for all natural integers $n$: $$u_{n+1} = \frac{1 + 0.5u_{n}}{0.5 + u_{n}}$$ It is admitted that all terms of this sequence are defined and strictly positive.
Consider the following algorithm:
Input
Let $n$ be a non-zero natural integer
Initialization
Assign to $u$ the value 2
Processing
FOR $i$ going from 1 to $n$
and
Assign to $u$ the value $\frac{1 + 0.5u}{0.5 + u}$
output
Display $u$
END FOR
Reproduce and complete the following table by running this algorithm for $n = 3$. The values of $u$ should be rounded to the nearest thousandth.
$i$
1
2
3
$u$
For $n = 12$, the previous table was extended and we obtained:
$i$
4
5
6
7
8
9
10
11
12
$u$
1.0083
0.9973
1.0009
0.9997
1.0001
0.99997
1.00001
0.999996
1.000001
Conjecture the behavior of the sequence $(u_{n})$ at infinity.
Consider the sequence $(v_{n})$ defined, for all natural integers $n$, by: $v_{n} = \frac{u_{n} - 1}{u_{n} + 1}$. a. Prove that the sequence $(v_{n})$ is geometric with common ratio $-\frac{1}{3}$. b. Calculate $v_{0}$ then write $v_{n}$ as a function of $n$.
a. Show that, for all natural integers $n$, we have: $v_{n} \neq 1$. b. Show that, for all natural integers $n$, we have: $u_{n} = \frac{1 + v_{n}}{1 - v_{n}}$. c. Determine the limit of the sequence $(u_{n})$.
\textbf{Exercise 4 (5 points) -- Candidates who have NOT chosen the specialization option}
\textbf{Part A}
Consider the sequence $(u_{n})$ defined by: $u_{0} = 2$ and, for all natural integers $n$:
$$u_{n+1} = \frac{1 + 3u_{n}}{3 + u_{n}}$$
It is admitted that all terms of this sequence are defined and strictly positive.
\begin{enumerate}
\item Prove by induction that, for all natural integers $n$, we have: $u_{n} > 1$.
\item a. Establish that, for all natural integers $n$, we have: $u_{n+1} - u_{n} = \frac{(1 - u_{n})(1 + u_{n})}{3 + u_{n}}$.\\
b. Determine the direction of variation of the sequence $(u_{n})$.\\
Deduce that the sequence $(u_{n})$ converges.
\end{enumerate}
\textbf{Part B}
Consider the sequence $(u_{n})$ defined by: $u_{0} = 2$ and, for all natural integers $n$:
$$u_{n+1} = \frac{1 + 0.5u_{n}}{0.5 + u_{n}}$$
It is admitted that all terms of this sequence are defined and strictly positive.
\begin{enumerate}
\item Consider the following algorithm:
\begin{center}
\begin{tabular}{ | l | l | }
\hline
Input & Let $n$ be a non-zero natural integer \\
\hline
Initialization & Assign to $u$ the value 2 \\
\hline
Processing & FOR $i$ going from 1 to $n$ \\
and & Assign to $u$ the value $\frac{1 + 0.5u}{0.5 + u}$ \\
output & Display $u$ \\
\hline
& END FOR \\
\hline
\end{tabular}
\end{center}
Reproduce and complete the following table by running this algorithm for $n = 3$. The values of $u$ should be rounded to the nearest thousandth.
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
$i$ & 1 & 2 & 3 \\
\hline
$u$ & & & \\
\hline
\end{tabular}
\end{center}
\item For $n = 12$, the previous table was extended and we obtained:
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | }
\hline
$i$ & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline
$u$ & 1.0083 & 0.9973 & 1.0009 & 0.9997 & 1.0001 & 0.99997 & 1.00001 & 0.999996 & 1.000001 \\
\hline
\end{tabular}
\end{center}
Conjecture the behavior of the sequence $(u_{n})$ at infinity.
\item Consider the sequence $(v_{n})$ defined, for all natural integers $n$, by: $v_{n} = \frac{u_{n} - 1}{u_{n} + 1}$.\\
a. Prove that the sequence $(v_{n})$ is geometric with common ratio $-\frac{1}{3}$.\\
b. Calculate $v_{0}$ then write $v_{n}$ as a function of $n$.
\item a. Show that, for all natural integers $n$, we have: $v_{n} \neq 1$.\\
b. Show that, for all natural integers $n$, we have: $u_{n} = \frac{1 + v_{n}}{1 - v_{n}}$.\\
c. Determine the limit of the sequence $(u_{n})$.
\end{enumerate}