3. Let $(v_n)$ be the sequence defined for every natural number $n$ by:
$$v_n = \frac{n}{2 + \cos(n)}.$$
Statement 3: The sequence $(v_n)$ diverges to $+\infty$.
4. In space with respect to an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider points $A(1; 1; 2), B(5; -1; 8)$ and $C(2; 1; 3)$. Statement 4: $\overrightarrow{AB} \cdot \overrightarrow{AC} = 10$ and a measure of angle $\widehat{BAC}$ is $30°$.

A team of biologists is studying the evolution of the area covered by a marine algae called seagrass, on the bottom of Alycastre Bay, near the island of Porquerolles. The studied area has a total area of 20 hectares (ha), and on July 1, 2024, the seagrass covered 1 ha of this area.
Part A: Study of a discrete model
For any natural integer $n$, we denote by $u _ { n }$ the area of the zone, in hectares, covered by seagrass on July 1 of the year $2024 + n$. Thus, $u _ { 0 } = 1$. A study conducted on this area made it possible to establish that for any natural integer $n$: $$u _ { n + 1 } = - 0{,}02 u _ { n } ^ { 2 } + 1{,}3 u _ { n }$$

1. Calculate the area that seagrass should cover on July 1, 2025 according to this model.
2. We denote by $h$ the function defined on $[0;20]$ by $$h ( x ) = - 0{,}02 x ^ { 2 } + 1{,}3 x$$ We admit that $h$ is increasing on $[0;20]$. a. Prove that for any natural integer $n$, $1 \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 20$. b. Deduce that the sequence $(u _ { n })$ converges. We denote by $L$ its limit. c. Justify that $L = 15$.
3. The biologists wish to know after how long the area covered by seagrass will exceed 14 hectares. a. Without any calculation, justify that, according to this model, this will occur. b. Copy and complete the following algorithm so that at the end of execution, it displays the answer to the biologists' question. \begin{verbatim} def seuil(): n=0 u= 1 while ...... : n=...... u=...... return n \end{verbatim}

Part B: Study of a continuous model
We wish to describe the area of the studied zone covered by seagrass over time with a continuous model. In this model, for a duration $t$, in years, elapsed from July 1, 2024, the area of the studied zone covered by seagrass is given by $f ( t )$, where $f$ is a function defined on $[ 0 ; + \infty [$ satisfying:

• $f ( 0 ) = 1$;
• $f$ does not vanish on $[ 0 ; + \infty [$;
• $f$ is differentiable on $[ 0 ; + \infty [$;
• $f$ is a solution on $[ 0 ; + \infty [$ of the differential equation $$\left( E _ { 1 } \right) : \quad y ^ { \prime } = 0{,}02 y ( 15 - y ) .$$

We admit that such a function $f$ exists; the purpose of this part is to determine an expression for it. We denote by $f ^ { \prime }$ the derivative function of $f$.

1. Let $g$ be the function defined on $\left[ 0 ; + \infty \left[ \text{ by } g ( t ) = \frac { 1 } { f ( t ) } \right. \right.$. Show that $g$ is a solution of the differential equation $$\left( E _ { 2 } \right) : \quad y ^ { \prime } = - 0{,}3 y + 0{,}02 .$$
2. Give the solutions of the differential equation $( E _ { 2 } )$.
3. Deduce that for all $t \in [ 0 ; + \infty [$: $$f ( t ) = \frac { 15 } { 14 \mathrm { e } ^ { - 0{,}3 t } + 1 }$$
4. Determine the limit of $f$ as $+ \infty$.
5. Solve in the interval $[ 0 ; + \infty [$ the inequality $f ( t ) > 14$. Interpret the result in the context of the exercise.

Exercise 3
For each of the following statements, indicate whether it is true or false. Justify each answer. An unjustified answer earns no points.

1. The sequence $(u_n)$ is defined for every natural integer $n$ by $$u_n = \frac{1 + 5^n}{2 + 3^n}$$ Statement 1: The sequence $(u_n)$ converges to $\frac{5}{3}$.
   
2. We consider the sequence $(w_n)$ defined by: $$w_0 = 0 \text{ and, for every natural integer } n,\ w_{n+1} = 3w_n - 2n + 3.$$ Statement 2: For every natural integer $n$, $w_n \geqslant n$.
   
3. We consider the function $f$ defined on $]0; +\infty[$ whose representative curve $\mathscr{C}_f$ is given in an orthonormal coordinate system in the figure (Fig. 1). We specify that:
   • $T$ is the tangent to $\mathscr{C}_f$ at point A with abscissa 8;
   • The x-axis is the horizontal tangent to $\mathscr{C}_f$ at the point with abscissa 1.
   Statement 3: According to the graph, the function $f$ is convex on its domain of definition.
   
4. Statement 4: For every real $x > 0$, $\ln(x) - x + 1 \leqslant 0$, where $\ln$ denotes the natural logarithm function.

The series $\sum _ { n = 1 } ^ { \infty } a _ { n }$ where $a _ { n } = ( - 1 ) ^ { n + 1 } n ^ { 4 } e ^ { - n ^ { 2 } }$
(a) has unbounded partial sums;
(b) is absolutely convergent;
(c) is convergent but not absolutely convergent;
(d) is not convergent, but partial sums oscillate between $-1$ and $+1$.

Let $f$ be continuously differentiable on $\mathbb { R }$. Let $f _ { n } ( x ) = n \left( f \left( x + \frac { 1 } { n } \right) - f ( x ) \right)$. Then,
(a) $f _ { n }$ converges uniformly on $\mathbb { R }$;
(b) $f _ { n }$ converges on $\mathbb { R }$, but not necessarily uniformly;
(c) $f _ { n }$ converges to the derivative of $f$ uniformly on $[ 0,1 ]$;
(d) there is no guarantee that $f _ { n }$ converges on any open interval.

Consider a sequence of polynomials with real coefficients defined by $$p_{0}(x) = \left(x^{2}+1\right)\left(x^{2}+2\right) \cdots\left(x^{2}+1009\right)$$ with subsequent polynomials defined by $p_{k+1}(x) := p_{k}(x+1) - p_{k}(x)$ for $k \geq 0$. Find the least $n$ such that $$p_{n}(1) = p_{n}(2) = \cdots = p_{n}(5000)$$

[12 points] Let $f$ be a function from natural numbers to natural numbers that satisfies
$$\begin{aligned} & f ( n ) = n - 2 \quad \text { for } n > 3000 \\ & f ( n ) = f ( f ( n + 5 ) ) \quad \text { for } n \leq 3000 \end{aligned}$$
Show that $f ( 2022 )$ is uniquely decided and find its value.

For a sequence $a _ { i }$ of real numbers, we say that $\sum a _ { i }$ converges if $\lim _ { n \rightarrow \infty } \left( \sum _ { i = 1 } ^ { n } a _ { i } \right)$ is finite. In this question all $a _ { i } > 0$.
Statements
(21) If $\sum a _ { i }$ converges, then $a _ { i } \rightarrow 0$ as $i \rightarrow \infty$. (22) If $a _ { i } < \frac { 1 } { i }$ for all $i$, then $\sum a _ { i }$ converges. (23) If $\sum a _ { i }$ converges, then $\sum ( - 1 ) ^ { i } a _ { i }$ also converges. (24) If $\sum a _ { i }$ does not converge, then $\sum i \tan \left( a _ { i } \right)$ cannot converge.

Two mighty frogs jump once per unit time on the number line as described in the question.
The second frog starts at $x=0$ and jumps $i+1$ steps to the right just after $t=i$, so that at times $t=0,1,2,3,\ldots$ this frog is at positions $x=0,1,3,6,\ldots$ respectively. How many numbers of the form $7n+1$ (with $n$ an integer) does the frog visit from $t=0$ to $t=99$ (both endpoints included)? [3 points]

What is the value of $\lim _ { n \rightarrow \infty } \left( \sqrt { n ^ { 2 } + 6 n + 4 } - n \right)$? [3 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) 1
(4) 2
(5) 3

For a sequence $\left\{ a _ { n } \right\}$ where the sum of the first $n$ terms $S _ { n } = 2 n + \frac { 1 } { 2 ^ { n } }$, what is the value of $\lim _ { n \rightarrow \infty } a _ { n }$? [3 points]
(1) 2
(2) 1
(3) $\frac { 1 } { 2 }$
(4) $\frac { 1 } { 4 }$
(5) 0

For a real number $a$ ($a > 1$), let $b = \sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { a } \right) ^ { n }$ be represented as in [Figure 1], and for a real number $c$, let $d = 16 ^ { c }$ be represented as in [Figure 2].
For the real numbers $x$, $y$, $z$ in the figure below, find the value of $\frac { x z } { y }$. [4 points]

When a sequence $\left\{ a _ { n } \right\}$ satisfies $n < a _ { n } < n + 1$ for all natural numbers $n$, what is the value of $\lim _ { n \rightarrow \infty } \frac { n ^ { 2 } } { a _ { 1 } + a _ { 2 } + \cdots + a _ { n } }$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Two sequences $\left\{ a _ { n } \right\} , \left\{ b _ { n } \right\}$ are given by $$\begin{aligned} & a _ { n } = \frac { 1 } { 2 ^ { n - 1 } } \cos \frac { ( n - 1 ) \pi } { 2 } \\ & b _ { n } = \frac { 1 + ( - 1 ) ^ { n - 1 } } { 2 ^ { n } } \end{aligned}$$ Which of the following in are correct? [4 points] 〈Remarks〉 ㄱ. For all natural numbers $k$, $a _ { 3 k } < 0$. ㄴ. For all natural numbers $k$, $a _ { 4 k - 1 } + b _ { 4 k - 1 } = 0$. ㄷ. $\sum _ { n = 1 } ^ { \infty } a _ { n } = \frac { 3 } { 5 } \sum _ { n = 1 } ^ { \infty } b _ { n }$
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄴ
(5) ㄴ, ㄷ

Find the value of $\lim _ { n \rightarrow \infty } \frac { 5 \cdot 3 ^ { n + 1 } - 2 ^ { n + 1 } } { 3 ^ { n } + 2 ^ { n } }$. [3 points]

For a natural number $p \geqq 2$, a sequence $\left\{ a _ { n } \right\}$ satisfies the following three conditions. Which of the following in are correct? [4 points] Conditions (가) $a _ { 1 } = 0$ (나) $a _ { k + 1 } = a _ { k } + 1 \quad ( 1 \leqq k \leqq p - 1 )$ (다) $a _ { k + p } = a _ { k } \quad ( k = 1,2,3 , \cdots )$ 〈Remarks〉 ㄱ. $a _ { 2 k } = 2 a _ { k }$ ㄴ. $a _ { 1 } + a _ { 2 } + \cdots + a _ { p } = \frac { p ( p - 1 ) } { 2 }$ ㄷ. $a _ { p } + a _ { 2 p } + \cdots + a _ { k p } = k ( p - 1 )$
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

What is the value of $\lim _ { n \rightarrow \infty } \frac { 3 + \left( \frac { 1 } { 3 } \right) ^ { n } } { 2 + \left( \frac { 1 } { 2 } \right) ^ { n } }$? [2 points]
(1) 1
(2) $\frac { 3 } { 2 }$
(3) 2
(4) $\frac { 5 } { 2 }$
(5) 3

For the sequence $\left\{ \left( \frac { 2 x - 1 } { 4 } \right) ^ { n } \right\}$ to converge, let $k$ be the number of integers $x$. Find the value of $10 k$. [3 points]

For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 0 and common difference not equal to 0, a sequence $\left\{ b _ { n } \right\}$ satisfies $a _ { n + 1 } b _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$. Find the value of $b _ { 27 }$. [4 points]

For a sequence $\left\{ a _ { n } \right\}$ with $a _ { 1 } = 2$ and $a _ { n + 1 } = 2 a _ { n } + 2$, what is the value of $a _ { 10 }$? [3 points]
(1) 1022
(2) 1024
(3) 2021
(4) 2046
(5) 2082

What is the value of $\lim _ { n \rightarrow \infty } \frac { n } { \sqrt { 4 n ^ { 2 } + 1 } + \sqrt { n ^ { 2 } + 2 } }$? [2 points]
(1) 1
(2) $\frac { 1 } { 2 }$
(3) $\frac { 1 } { 3 }$
(4) $\frac { 1 } { 4 }$
(5) $\frac { 1 } { 5 }$

For a sequence $\left\{ a _ { n } \right\}$ with $\sum _ { n = 1 } ^ { \infty } \frac { a _ { n } } { 4 ^ { n } } = 2$, find the value of $\lim _ { n \rightarrow \infty } \frac { a _ { n } + 4 ^ { n + 1 } - 3 ^ { n - 1 } } { 4 ^ { n - 1 } + 3 ^ { n + 1 } }$. [3 points]

What is the value of $\lim _ { n \rightarrow \infty } \frac { 2 } { \sqrt { n ^ { 2 } + 2 n } - \sqrt { n ^ { 2 } + 1 } }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Let $a_n$ denote the sum of all natural numbers such that when divided by a natural number $n$ ($n \geqq 2$), the quotient and remainder are equal. For example, when divided by 4, the natural numbers for which the quotient and remainder are equal are $5, 10, 15$, so $a_4 = 5 + 10 + 15 = 30$. Find the minimum value of the natural number $n$ satisfying $a_n > 500$. [4 points]

What is the value of $\lim _ { n \rightarrow \infty } \frac { ( n + 1 ) ( 3 n - 1 ) } { 2 n ^ { 2 } + 1 }$? [2 points]
(1) $\frac { 3 } { 2 }$
(2) 2
(3) $\frac { 5 } { 2 }$
(4) 3
(5) $\frac { 7 } { 2 }$