Exercise 4 — Candidates who have not chosen the specialization option
Let the numerical sequence ( $u _ { n }$ ) defined on the set of natural integers $\mathbb { N }$ by $$\left\{ \begin{aligned}
u _ { 0 } & = 2 \\
\text { and for all natural integer } n , u _ { n + 1 } & = \frac { 1 } { 5 } u _ { n } + 3 \times 0{,}5 ^ { n } .
\end{aligned} \right.$$
a. Copy and, using a calculator, complete the table of values of the sequence $\left( u _ { n } \right)$ approximated to $10 ^ { - 2 }$ near:
$n$
0
1
2
3
4
5
6
7
8
$u _ { n }$
2
b. Based on this table, state a conjecture about the direction of variation of the sequence $\left( u _ { n } \right)$.
a. Prove, by induction, that for all non-zero natural integer $n$ we have $$u _ { n } \geqslant \frac { 15 } { 4 } \times 0{,}5 ^ { n }$$ b. Deduce that, for all natural integer $n$ non-zero, $u _ { n + 1 } - u _ { n } \leqslant 0$. c. Prove that the sequence ( $u _ { n }$ ) is convergent.
We propose, in this question, to determine the limit of the sequence $\left( u _ { n } \right)$. Let $\left( v _ { n } \right)$ be the sequence defined on $\mathbb { N }$ by $v _ { n } = u _ { n } - 10 \times 0{,}5 ^ { n }$. a. Prove that the sequence $\left( v _ { n } \right)$ is a geometric sequence with ratio $\frac { 1 } { 5 }$. We will specify the first term of the sequence $\left( v _ { n } \right)$. b. Deduce that for all natural integer $n$, $$u _ { n } = - 8 \times \left( \frac { 1 } { 5 } \right) ^ { n } + 10 \times 0{,}5 ^ { n }.$$ c. Determine the limit of the sequence ( $u _ { n }$ ).
Copy and complete lines (1), (2) and (3) of the following algorithm, so that it displays the smallest value of $n$ such that $u _ { n } \leqslant 0{,}01$.
Input:
$n$ and $u$ are numbers
Initialization :
$n$ takes the value 0
$u$ takes the value 2
Processing :
While $\ldots$
(1)
$n$ takes the value $\ldots$
(2)
$u$ takes the value $\ldots$
(3)
End While
Output:
Display $n$
\section*{Exercise 4 — Candidates who have not chosen the specialization option}
Let the numerical sequence ( $u _ { n }$ ) defined on the set of natural integers $\mathbb { N }$ by
$$\left\{ \begin{aligned}
u _ { 0 } & = 2 \\
\text { and for all natural integer } n , u _ { n + 1 } & = \frac { 1 } { 5 } u _ { n } + 3 \times 0{,}5 ^ { n } .
\end{aligned} \right.$$
\begin{enumerate}
\item a. Copy and, using a calculator, complete the table of values of the sequence $\left( u _ { n } \right)$ approximated to $10 ^ { - 2 }$ near:
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | }
\hline
$n$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
$u _ { n }$ & 2 & & & & & & & & \\
\hline
\end{tabular}
\end{center}
b. Based on this table, state a conjecture about the direction of variation of the sequence $\left( u _ { n } \right)$.
\item a. Prove, by induction, that for all non-zero natural integer $n$ we have
$$u _ { n } \geqslant \frac { 15 } { 4 } \times 0{,}5 ^ { n }$$
b. Deduce that, for all natural integer $n$ non-zero, $u _ { n + 1 } - u _ { n } \leqslant 0$.\\
c. Prove that the sequence ( $u _ { n }$ ) is convergent.
\item We propose, in this question, to determine the limit of the sequence $\left( u _ { n } \right)$. Let $\left( v _ { n } \right)$ be the sequence defined on $\mathbb { N }$ by $v _ { n } = u _ { n } - 10 \times 0{,}5 ^ { n }$.\\
a. Prove that the sequence $\left( v _ { n } \right)$ is a geometric sequence with ratio $\frac { 1 } { 5 }$. We will specify the first term of the sequence $\left( v _ { n } \right)$.\\
b. Deduce that for all natural integer $n$,
$$u _ { n } = - 8 \times \left( \frac { 1 } { 5 } \right) ^ { n } + 10 \times 0{,}5 ^ { n }.$$
c. Determine the limit of the sequence ( $u _ { n }$ ).
\item Copy and complete lines (1), (2) and (3) of the following algorithm, so that it displays the smallest value of $n$ such that $u _ { n } \leqslant 0{,}01$.
\begin{center}
\begin{tabular}{ | l l r | }
\hline
Input: & $n$ and $u$ are numbers & \\
Initialization : & $n$ takes the value 0 & \\
& $u$ takes the value 2 & \\
Processing : & While $\ldots$ & (1) \\
& $n$ takes the value $\ldots$ & (2) \\
& $u$ takes the value $\ldots$ & (3) \\
& End While & \\
Output: & Display $n$ & \\
\hline
\end{tabular}
\end{center}
\end{enumerate}