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
$$\left\{ \begin{aligned} u _ { 0 } & = 2 \\ \text { and for all natural integer } n , u _ { n + 1 } & = \frac { 1 } { 5 } u _ { n } + 3 \times 0{,}5 ^ { n } . \end{aligned} \right.$$

1. 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 $\left( u _ { n } \right)$.
   
2. 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.
   
3. 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 ( $u _ { n }$ ).
   
4. 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 $n$

   
   

We denote $V_n(z) = U_{n+1}(z, -1)$ for all $z \in \mathbb{C}$ and $n \in \mathbb{N}$, where $U(a,b) = (U_n(a,b))_{n \in \mathbb{N}}$ is the unique sequence satisfying $E_{a,b}$ with initial conditions $U_0(a,b) = 0$ and $U_1(a,b) = 1$. Show that, for all $z \in \mathbb{C}$ and $n \in \mathbb{N}$, we have $$V_n(z) = \sum_{j=0}^{\lfloor n/2 \rfloor} \binom{n-j}{j} (2z)^{n-2j} (-1)^j$$ One may proceed by induction.

Verify that, if $\left( z _ { k } \right)$ is $p$-periodic, then $\forall n \in \mathbb { N } , \forall k \in \mathbb { N } , z _ { n + k p } = z _ { n }$.

We assume that $p$ is an integer greater than or equal to 2, that $\left( a _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( b _ { k } \right) _ { k \in \mathbb { N } }$ are two sequences of real numbers that are $p$-periodic and that $\forall k \in \mathbb { N } , b _ { k } \neq 0$. We denote by Sol(II.2) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ that satisfy the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad b _ { k } z _ { k + 1 } + a _ { k } z _ { k } + b _ { k - 1 } z _ { k - 1 } = 0$$ We fix $\left( y _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( z _ { k } \right) _ { k \in \mathbb { N } }$, two solution sequences of (II.2). We set for all $k \in \mathbb { N } , W _ { k } = b _ { k } \left( y _ { k } z _ { k + 1 } - z _ { k } y _ { k + 1 } \right)$. Show that the sequence $\left( W _ { k } \right) _ { k \in \mathbb { N } }$ is constant.

We assume that $p$ is an integer greater than or equal to 2, that $\left( a _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( b _ { k } \right) _ { k \in \mathbb { N } }$ are two sequences of real numbers that are $p$-periodic and that $\forall k \in \mathbb { N } , b _ { k } \neq 0$. We denote by Sol(II.2) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ that satisfy the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad b _ { k } z _ { k + 1 } + a _ { k } z _ { k } + b _ { k - 1 } z _ { k - 1 } = 0$$ We fix $\left( y _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( z _ { k } \right) _ { k \in \mathbb { N } }$, two solution sequences of (II.2), and set for all $k \in \mathbb { N } , W _ { k } = b _ { k } \left( y _ { k } z _ { k + 1 } - z _ { k } y _ { k + 1 } \right)$. Show that the two sequences $\left( y _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ form a basis of $\operatorname { Sol } ($ II.2 $)$ if and only if $W _ { 0 } \neq 0$.

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a differentiable function such that its derivative $f ^ { \prime }$ is a continuous function. Moreover, assume that for all $x \in \mathbb { R }$, $$0 \leq \left| f ^ { \prime } ( x ) \right| \leq \frac { 1 } { 2 }$$ Define a sequence of real numbers $\left\{ a _ { n } \right\} _ { n \in \mathbb { N } }$ by: $$\begin{gathered} a _ { 1 } = 1 , \\ a _ { n + 1 } = f \left( a _ { n } \right) \text { for all } n \in \mathbb { N } . \end{gathered}$$ Prove that there exists a positive real number $M$ such that for all $n \in \mathbb { N }$, $$\left| a _ { n } \right| \leq M$$

Let $\left\{ u _ { n } \right\} _ { n \geq 1 }$ be a sequence of real numbers defined as $u _ { 1 } = 1$ and
$$u _ { n + 1 } = u _ { n } + \frac { 1 } { u _ { n } } \text { for all } n \geq 1 .$$
Prove that $u _ { n } \leq \frac { 3 \sqrt { n } } { 2 }$ for all $n$.