Consider the sequence $(u_n)$ defined by: $u_0 = 1$ and, for every natural integer $n$, $$u_{n+1} = \frac{4u_n}{u_n + 4}.$$ 1. The screenshot below presents the values, calculated using a spreadsheet, of the terms of the sequence $(u_n)$ for $n$ varying from 0 to 12, as well as those of the quotient $\frac{4}{u_n}$, (with, for the values of $u_n$, display of two digits for the decimal parts). Using these values, conjecture the expression of $\frac{4}{u_n}$ as a function of $n$. The purpose of this exercise is to prove this conjecture (question 5.) and to deduce the limit of the sequence $(u_n)$ (question 6.).
$n$
$u_n$
$\frac{4}{u_n}$
0
1,00
4
1
0,80
5
2
0,67
6
3
0,57
7
4
0,50
8
5
0,44
9
6
0,40
10
7
0,36
11
8
0,33
12
9
0,31
13
10
0,29
14
11
0,27
15
12
0,25
16
Prove by induction that, for every natural integer $n$, we have: $u_n > 0$.
Prove that the sequence $(u_n)$ is decreasing.
What can be concluded from questions 2. and 3. concerning the sequence $(u_n)$?
Consider the sequence $(v_n)$ defined for every natural integer $n$ by: $v_n = \frac{4}{u_n}$.
Prove that $(v_n)$ is an arithmetic sequence. Specify its common difference and its first term. Deduce, for every natural integer $n$, the expression of $v_n$ as a function of $n$. 6. Determine, for every natural integer $n$, the expression of $u_n$ as a function of $n$. Deduce the limit of the sequence $(u_n)$.
Consider the sequence $(u_n)$ defined by: $u_0 = 1$ and, for every natural integer $n$,
$$u_{n+1} = \frac{4u_n}{u_n + 4}.$$
1. The screenshot below presents the values, calculated using a spreadsheet, of the terms of the sequence $(u_n)$ for $n$ varying from 0 to 12, as well as those of the quotient $\frac{4}{u_n}$, (with, for the values of $u_n$, display of two digits for the decimal parts).\\
Using these values, conjecture the expression of $\frac{4}{u_n}$ as a function of $n$.\\
The purpose of this exercise is to prove this conjecture (question 5.) and to deduce the limit of the sequence $(u_n)$ (question 6.).
\begin{center}
\begin{tabular}{ | c | c | c | }
\hline
$n$ & $u_n$ & $\frac{4}{u_n}$ \\
\hline
0 & 1,00 & 4 \\
\hline
1 & 0,80 & 5 \\
\hline
2 & 0,67 & 6 \\
\hline
3 & 0,57 & 7 \\
\hline
4 & 0,50 & 8 \\
\hline
5 & 0,44 & 9 \\
\hline
6 & 0,40 & 10 \\
\hline
7 & 0,36 & 11 \\
\hline
8 & 0,33 & 12 \\
\hline
9 & 0,31 & 13 \\
\hline
10 & 0,29 & 14 \\
\hline
11 & 0,27 & 15 \\
\hline
12 & 0,25 & 16 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}
\setcounter{enumi}{1}
\item Prove by induction that, for every natural integer $n$, we have: $u_n > 0$.
\item Prove that the sequence $(u_n)$ is decreasing.
\item What can be concluded from questions 2. and 3. concerning the sequence $(u_n)$?
\item Consider the sequence $(v_n)$ defined for every natural integer $n$ by: $v_n = \frac{4}{u_n}$.
\end{enumerate}
Prove that $(v_n)$ is an arithmetic sequence.\\
Specify its common difference and its first term.\\
Deduce, for every natural integer $n$, the expression of $v_n$ as a function of $n$.
6. Determine, for every natural integer $n$, the expression of $u_n$ as a function of $n$.
Deduce the limit of the sequence $(u_n)$.