5. Using the previous questions, the following result can be established, which is admitted.
For every non-zero natural integer $n$,
$$A ^ { n } = \left( \begin{array} { c c } - 2 ^ { n + 1 } + 3 ^ { n + 1 } & 3 \times 2 ^ { n + 1 } - 2 \times 3 ^ { n + 1 } \\ - 2 ^ { n } + 3 ^ { n } & 3 \times 2 ^ { n } - 2 \times 3 ^ { n } \end{array} \right)$$
Deduce an expression for $u _ { n }$ as a function of $n$. Does the sequence ( $u _ { n }$ ) have a limit?

APPENDIX for EXERCISE 3, to be returned with the answer sheet

Graphical representation $\mathscr { C } _ { 1 }$ of the function $f _ { 1 }$ [Figure]

The sequence ( $u _ { n }$ ) is defined by:
$$u _ { 0 } = 0 \quad \text { and, for all natural integer } n , u _ { n + 1 } = \frac { 1 } { 2 - u _ { n } } .$$

1. a. Using the calculation of the first terms of the sequence ( $u _ { n }$ ), conjecture the explicit form of $u _ { n }$ as a function of $n$. Prove this conjecture. b. Deduce the value of the limit $\ell$ of the sequence $\left( u _ { n } \right)$.
2. Complete, in appendix 2, the algorithm to determine the value of the smallest integer $n$ such that $\left| u _ { n + 1 } - u _ { n } \right| \leqslant 10 ^ { - 3 }$.

Let $\left(u_n\right)$ be the sequence defined by $u_0 = 1$ and for every natural integer $n$ $$u_{n+1} = \frac{u_n}{1 + u_n}$$

1. a. Calculate the terms $u_1, u_2$ and $u_3$. Give the results as irreducible fractions. b. Copy the Python script below and complete lines 3 and 6 so that \texttt{liste($k$)} takes as parameter a natural integer $k$ and returns the list of the first values of the sequence $\left(u_n\right)$ from $u_0$ to $u_k$. \begin{verbatim} def liste(k) : L = [] u=... for i in range(0, k+1) : L.append(u) u = ... return(L) \end{verbatim}
   
2. It is admitted that, for every natural integer $n$, $u_n$ is strictly positive. Determine the direction of variation of the sequence $(u_n)$.
3. Deduce that the sequence $(u_n)$ converges.
4. Determine the value of its limit.
5. a. Conjecture an expression of $u_n$ as a function of $n$. b. Prove by induction the previous conjecture.

Exercise 4 — 6 points Theme: sequences, functions Let $(u_n)$ be the sequence defined by $u_0 = -1$ and, for every natural number $n$: $$u_{n+1} = 0.9u_n - 0.3.$$

1. a. Prove by induction that, for all $n \in \mathbb{N}$, $u_n = 2 \times 0.9^n - 3$. b. Deduce that for all $n \in \mathbb{N}$, $-3 < u_n \leq -1$. c. Prove that the sequence $(u_n)$ is strictly decreasing. d. Prove that the sequence $(u_n)$ converges and specify its limit.
2. We propose to study the function $g$ defined on $]-3; -1]$ by: $$g(x) = \ln(0.5x + 1.5) - x.$$ a. Justify all the information given by the variations table of function $g$ (limits, variations, image of $-1$). b. Deduce that the equation $g(x) = 0$ has exactly one solution which we will denote $\alpha$ and for which we will give an interval of amplitude $10^{-3}$.
3. In the rest of the exercise, we consider the sequence $(v_n)$ defined for all $n \in \mathbb{N}$ by: $$v_n = \ln(0.5u_n + 1.5).$$ a. Using the formula given in question 1.a., prove that the sequence $v$ is arithmetic with common difference $\ln(0.9)$. b. Let $n$ be a natural number. Prove that $u_n = v_n$ if and only if $g(u_n) = 0$. c. Prove that there is no rank $k \in \mathbb{N}$ for which $u_k = \alpha$. d. Deduce that there is no rank $k \in \mathbb{N}$ for which $v_k = u_k$.

Let $a _ { 0 }$ and $a _ { 1 }$ be complex numbers and define $a _ { n } = 2 a _ { n - 1 } + a _ { n - 2 }$ for $n \geq 2$.
(A) Show that there are polynomials $p ( z ) , q ( z ) \in \mathbb { C } [ z ]$ such that $q ( 0 ) \neq 0$ and $\sum _ { n \geq 0 } a _ { n } z ^ { n }$ is the Taylor series expansion (around 0) of $\frac { p ( z ) } { q ( z ) }$.
(B) Let $a _ { 0 } = 1$ and $a _ { 1 } = 2$. Show that there exist complex numbers $\beta _ { 1 } , \beta _ { 2 } , \gamma _ { 1 } , \gamma _ { 2 }$ such that $$a _ { n } = \beta _ { 1 } \gamma _ { 1 } ^ { n + 1 } + \beta _ { 2 } \gamma _ { 2 } ^ { n + 1 }$$ for all $n$.

19. (This question is worth 13 points) Let $S _ { n }$ be the sum of the first $n$ terms of the sequence $\left\{ a _ { n } \right\}$. Given $a _ { 1 } = 1 , a _ { 2 } = 2$, and $a _ { n + 2 } = 3 S _ { n } - S _ { n + 1 } , n \in \mathbb { N } ^ { * }$. (I) Prove that: $a _ { n + 2 } = 3 a _ { n }$ (II) Find $\mathrm { S } _ { \mathrm { n } }$

13. Let $s _ { n }$ be the sum of the first n terms of the geometric sequence $\left\{ a _ { n } \right\}$. If $a _ { 1 } = 1$ and $3 s _ { 1 } , 2 s _ { 2 } , s _ { 3 }$ form an arithmetic sequence, then $a _ { n } = $ $\_\_\_\_$.

Let $\mathrm { S } _ { \mathrm { n } }$ be the sum of the first $n$ terms of sequence $\left\{ \mathrm { a } _ { \mathrm { n } } \right\}$, and $a _ { 1 } = - 1 , a _ { \mathrm { n } + 1 } = S _ { n } S _ { n + 1 }$. Then $S _ { n } = $ $\_\_\_\_$ .

16. (This question is worth 12 points) Let the sequence $\left\{ a _ { n } \right\} ( n = 1,2,3 \ldots )$ have the sum of the first $n$ terms $S _ { n } = 2 a _ { n } - a _ { 1 }$, and $a _ { 1 } , a _ { 1 } + 1 , a _ { 3 }$ form an arithmetic sequence. (1) Find the general term formula of the sequence $\left\{ a _ { n } \right\}$; (2) Let $T _ { n }$ be the sum of the first $n$ terms of the sequence $\left\{ \frac { 1 } { a _ { n } } \right\}$. Find $T _ { n }$.

16. Let the sequence $\left\{ a _ { n } \right\}$ have the sum of the first $n$ terms $S _ { n } = 2 a _ { n } - a _ { 3 }$, and $a _ { 1 }$, $a _ { 2 } + 1$, $a _ { 3 }$ form an arithmetic sequence.
(1) Find the general term formula of the sequence $\left\{ a _ { n } \right\}$;
(2) Let $T _ { n }$ be the sum of the first $n$ terms of the sequence $\left\{ \frac { 1 } { a _ { n } } \right\}$. Find the minimum value of $n$ such that $\left| T _ { n } - 1 \right| < \frac { 1 } { 1000 }$.

Given the sequence $\{a_n\}$ satisfies $a_{n+2} = qa_n$ (where q is a real number and $q \neq 1$), $n \in \mathbb{N}^*$, $a_1 = 1$, $a_2 = 2$, and $a_2 + a_3$, $a_3 + a_4$, $a_4 + a_5$ form an arithmetic sequence.
(I) Find the value of q and the general term formula of $\{a_n\}$;
(II) Let $b_n = \frac{\log_2 a_{2n}}{a_{2n-1}}$, $n \in \mathbb{N}^*$. Find the sum of the first n terms of the sequence $\{b_n\}$.

17. (10 points) Let $S _ { n }$ denote the sum of the first $n$ terms of the sequence $\left\{ a _ { n } \right\}$. Given that $a _ { 1 } = 1$ and $\left\{ \frac { S _ { n } } { a _ { n } } \right\}$ is an arithmetic sequence with common difference $\frac { 1 } { 3 }$.
(1) Find the general term formula for $\left\{ a _ { n } \right\}$;
(2) Prove that $\frac { 1 } { a _ { 1 } } + \frac { 1 } { a _ { 2 } } + \cdots + \frac { 1 } { a _ { n } } < 2$ .

In the sequence $\left\{ a_{n} \right\}$ , $a_{2} = 1$ . Let $S_{n}$ be the sum of the first $n$ terms of $\left\{ a_{n} \right\}$ . $2S_{n} = na_{n}$ .
(1) Find the general term formula for $\left\{ a_{n} \right\}$ ;
(2) Find the sum $T_{n}$ of the first $n$ terms of the sequence $\left\{ \frac{a_{n} + 1}{2^{n}} \right\}$ .

We consider the sequence $x = (x_n)_{n \geqslant 0}$ defined by $$x_0 = 1, \quad x_1 = 1, \quad x_2 = 1, \quad x_3 = 0, \quad \text{and} \quad \forall n \in \mathbb{N}, x_{n+4} = x_{n+3} - 2x_{n+1}$$
We decide to modify only the value of $x_0$, by setting this time $x_0 = \frac{1}{2}$.
With this modification, quickly redo the study of questions II.C.2 and II.C.3.

Let $a$ and $b$ be two complex numbers such that $(a,b) \neq (0,0)$. We say that a complex sequence $U = (u_n)_{n \in \mathbb{N}}$ satisfies the recurrence relation $(E_{a,b})$ if $$\forall n \in \mathbb{N}, \quad u_{n+2} = 2a u_{n+1} + b u_n$$ We assume that $a^2 + b \neq 0$. We denote $d = R(a^2 + b)$. We call $W$ the sequence $W = ((a+d)^n)_{n \in \mathbb{N}}$ and $W'$ the sequence $W' = ((a-d)^n)_{n \in \mathbb{N}}$. Show that $U$ satisfies $E_{a,b}$ if and only if $U \in \operatorname{Vect}(W, W')$. Determine $U$ satisfying $E_{a,b}$ and the initial conditions $u_0 = 0$ and $u_1 = 1$, as a function of $d$, $W$ and $W'$.

Let $a$ and $b$ be two complex numbers such that $(a,b) \neq (0,0)$. We say that a complex sequence $U = (u_n)_{n \in \mathbb{N}}$ satisfies the recurrence relation $(E_{a,b})$ if $$\forall n \in \mathbb{N}, \quad u_{n+2} = 2a u_{n+1} + b u_n$$ We assume that $a^2 + b = 0$ and $a \neq 0$. We denote $W$ and $W'$ the sequences $W = (a^n)_{n \in \mathbb{N}}$ and $W' = (na^n)_{n \in \mathbb{N}}$. Show that $U$ satisfies $E_{a,b}$ if and only if $U \in \operatorname{Vect}(W, W')$. Determine $U$ satisfying $E_{a,b}$ and the initial conditions $u_0 = 0$ and $u_1 = 1$, as a function of $a$, $W$ and $W'$.

We assume $\alpha = 1$ and use the notation $V_n(z) = U_{n+1}(z,-1)$. Using the expansion of $H_t$ as a power series, deduce that $$\forall n \in \mathbb{N},\, \forall t \in ]0,\pi[, \quad V_n(\cos t) = \frac{\sin((n+1)t)}{\sin t}$$

In this subsection II.A, $a$ is a nonzero real number. We denote by Sol(II.1) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ satisfying the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad z _ { k + 1 } + a z _ { k } + z _ { k - 1 } = 0$$ Give the general form of sequences belonging to $\operatorname { Sol } ($ II.1 $)$ as a function of the complex roots $r _ { 1 }$ and $r _ { 2 }$ of the equation $r ^ { 2 } + a r + 1 = 0$. What are $r _ { 1 } + r _ { 2 }$ and $r _ { 1 } r _ { 2 }$?

For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convention $C_{0} = 1$. For every $x \in ]-\frac{1}{4}, \frac{1}{4}[$, we set $F(x) = \sum_{k=0}^{+\infty} C_{k} x^{k}$.
Determine, for every $x \in ]-\frac{1}{4}, \frac{1}{4}[$, an expression of $F(x)$ as a function of $x$.

For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convenience $C_{0} = 1$. For every $x \in \left]-\frac{1}{4}, \frac{1}{4}\right[$, we set $F(x) = \sum_{k=0}^{+\infty} C_{k} x^{k}$.
Determine, for every $x \in \left]-\frac{1}{4}, \frac{1}{4}\right[$, an expression of $F(x)$ as a function of $x$.

We set, for all $t \in I$, $$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$ Deduce that there exists a function $\varepsilon : I \rightarrow \{ - 1,1 \}$ such that $$\forall t \in I , \quad g ( t ) = 1 + \varepsilon ( t ) \sqrt { 1 - 4 t } .$$

Deduce from this, for every natural integer $k$, an expression for $P ^ { ( k ) }$ as a function of $T$, $k$ and $P ^ { ( 0 ) }$.

Let $\{x_n\}$ be a sequence such that $x_1 = 2$, $x_2 = 1$ and $2x_n - 3x_{n-1} + x_{n-2} = 0$ for $n > 2$. Find an expression for $x_n$.

Let $f(n)$ satisfy the recurrence $f(n) + f(n-1) = nf(n-1) + (n-1)f(n-2)$ with $f(0) = 1$, $f(1) = 0$. Find a closed form for $f(n)$.

Let $\{a_n\}_{n=0}^{\infty}$ be a sequence such that $a_0 = a_1 = 0$ and $a_{n+2} = 3a_{n+1} - 2a_n + 1, \forall n \geq 0$. Then $a_{25}a_{23} - 2a_{25}a_{22} - 2a_{23}a_{24} + 4a_{22}a_{24}$ is equal to
(1) 483
(2) 528
(3) 575
(4) 624