We define the sequences $(u_n)$ and $(v_n)$ on the set $\mathbb{N}$ of natural numbers by: $$u_0 = 0 ; v_0 = 1, \text{ and } \left\{\begin{array}{l} u_{n+1} = \dfrac{u_n + v_n}{2} \\ v_{n+1} = \dfrac{u_n + 2v_n}{3} \end{array}\right.$$
The purpose of this exercise is to study the convergence of sequences $(u_n)$ and $(v_n)$.

1. Calculate $u_1$ and $v_1$.
2. We consider the following algorithm:
   Variables: $u$, $v$ and $w$ real numbers; $N$ and $k$ integers Initialization: $u$ takes the value 0; $v$ takes the value 1 Start of algorithm Enter the value of $N$ For $k$ varying from 1 to $N$ $w$ takes the value $u$ $u$ takes the value $\dfrac{w + v}{2}$ $v$ takes the value $\dfrac{w + 2v}{3}$ End of For Display $u$ Display $v$ End of algorithm
   a. We execute this algorithm by entering $N = 2$. Copy and complete the table given below containing the state of variables during the execution of the algorithm.
   

   $k$	$w$	$u$	$v$
   1
   2

   
   b. For a given number $N$, what do the values displayed by the algorithm correspond to with respect to the situation studied in this exercise?
   
3. For all natural numbers $n$ we define the column vector $X_n$ by $X_n = \binom{u_n}{v_n}$ and the matrix $A$ by $$A = \left(\begin{array}{ll} \dfrac{1}{2} & \dfrac{1}{2} \\ \dfrac{1}{3} & \dfrac{2}{3} \end{array}\right).$$ a. Verify that, for all natural numbers $n$, $X_{n+1} = A X_n$. b. Prove by induction that $X_n = A^n X_0$ for all natural numbers $n$.
   
4. We define matrices $P$, $P'$ and $B$ by $$P = \left(\begin{array}{cc} \dfrac{4}{5} & \dfrac{6}{5} \\ -\dfrac{6}{5} & \dfrac{6}{5} \end{array}\right), \quad P' = \left(\begin{array}{cc} \dfrac{1}{2} & -\dfrac{1}{2} \\ \dfrac{1}{2} & \dfrac{1}{3} \end{array}\right)$$

The complex plane is equipped with an orthonormal coordinate system ($\mathrm{O}; \vec{u}, \vec{v}$). We consider the line $\mathscr{D}$ with equation $$7x - 3y - 1 = 0$$ We define the sequence $(A_{n})$ of points in the plane with coordinates $(x_{n}; y_{n})$ satisfying for all natural integer $n$: $$\left\{ \begin{array}{l} x_{0} = 1 \\ y_{0} = 2 \end{array} \quad \text{and} \quad \left\{ \begin{array}{l} x_{n+1} = -\frac{13}{2} x_{n} + 3 y_{n} \\ y_{n+1} = -\frac{35}{2} x_{n} + 8 y_{n} \end{array} \right. \right.$$
We denote by $M$ the matrix $\left( \begin{array}{cc} \frac{-13}{2} & 3 \\ \frac{-35}{2} & 8 \end{array} \right)$.

(Candidates who have followed the specialization course)
We observe the size of an ant colony every day. For any non-zero natural number $n$, we denote by $u _ { n }$ the number of ants, expressed in thousands, in this population at the end of the $n$-th day. At the beginning of the study the colony has 5000 ants and after one day it has 5100 ants. Thus, we have $u _ { 0 } = 5$ and $u _ { 1 } = 5.1$. We assume that the increase in the size of the colony from one day to the next decreases by $10\%$ each day. In other words, for any natural number $n$,
$$u _ { n + 2 } - u _ { n + 1 } = 0.9 \left( u _ { n + 1 } - u _ { n } \right) .$$

1. Prove that under these conditions, $u _ { 2 } = 5.19$.
2. For any natural number $n$, we set $V _ { n } = \binom { u _ { n + 1 } } { u _ { n } }$ and $A = \left( \begin{array} { c c } 1.9 & - 0.9 \\ 1 & 0 \end{array} \right)$. a. Prove that, for any natural number $n$, we have $V _ { n + 1 } = A V _ { n }$.

We then admit that, for any natural number $n$, $V _ { n } = A ^ { n } V _ { 0 }$. b. We set $P = \left( \begin{array} { c c } 0.9 & 1 \\ 1 & 1 \end{array} \right)$. We admit that the matrix $P$ is invertible.
Using a calculator, determine the matrix $P ^ { - 1 }$. By detailing the calculations, determine the matrix $D$ defined by $D = P ^ { - 1 } A P$. c. Prove by induction that, for any natural number $n$, we have $A ^ { n } = P D ^ { n } P ^ { - 1 }$. For any natural number $n$, we admit that
$$A ^ { n } = \left( \begin{array} { c c } - 10 \times 0.9 ^ { n + 1 } + 10 & 10 \times 0.9 ^ { n + 1 } - 9 \\ - 10 \times 0.9 ^ { n } + 10 & 10 \times 0.9 ^ { n } - 9 \end{array} \right) .$$
d. Deduce that, for any natural number $n$, $u _ { n } = 6 - 0.9 ^ { n }$.

1. Calculate the size of the colony at the end of the $10 ^ { \mathrm { th } }$ day. Round the result to the nearest ant.
2. Calculate the limit of the sequence $(u _ { n })$. Interpret this result in context.

The tribonacci numbers $\left\{ T _ { n } \right\}$ are defined for non-negative integers $n$ as follows.
$$\left\{ \begin{array} { l } T _ { 0 } = T _ { 1 } = 0 \\ T _ { 2 } = 1 \\ T _ { n + 3 } = T _ { n + 2 } + T _ { n + 1 } + T _ { n } \quad ( n \geq 0 ) \end{array} \right.$$
Answer the following questions.
(1) Find the matrix $A$ that satisfies Eq. (1.1) for all non-negative integers $n$.
$$\left( \begin{array} { l } T _ { n + 3 } \\ T _ { n + 2 } \\ T _ { n + 1 } \end{array} \right) = A \left( \begin{array} { l } T _ { n + 2 } \\ T _ { n + 1 } \\ T _ { n } \end{array} \right)$$
(2) Find the rank and the characteristic equation, i.e., the equation that eigenvalues satisfy, of the matrix $A$.
(3) Let $\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 }$ denote the eigenvalues of the matrix $A$. Express an eigenvector corresponding to each of the eigenvalues using $\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 }$.
(4) Prove that the matrix $A$ has only one real number eigenvalue. Letting $\lambda _ { 1 }$ correspond to this eigenvalue, prove that $1 < \lambda _ { 1 } < 2$.
(5) Prove that $T _ { n }$ can be expressed as $T _ { n } = c _ { 1 } \lambda _ { 1 } ^ { n } + c _ { 2 } \lambda _ { 2 } ^ { n } + c _ { 3 } \lambda _ { 3 } ^ { n }$ using constant complex numbers $c _ { 1 } , c _ { 2 } , c _ { 3 }$. You do not need to find values of $c _ { 1 } , c _ { 2 } , c _ { 3 }$ explicitly.
(6) Prove $\lim _ { n \rightarrow \infty } \frac { T _ { n + 1 } } { T _ { n } } = \lambda _ { 1 }$.