For questions 3. and 4., consider the sequence $(u_n)$ defined on $\mathbb{N}$ by: $$u_0 = 15 \text{ and for every natural number } n : u_{n+1} = 1{,}2\, u_n + 12.$$
Consider the sequence $(v_n)$ defined on $\mathbb{N}$ by: $v_n = u_n + 60$.
The sequence $(v_n)$ is: a. a decreasing sequence; b. a geometric sequence with common ratio 1,2; c. an arithmetic sequence with common difference 60; d. a sequence that is neither geometric nor arithmetic.

Léa spends a good part of her days playing a video game and is interested in the chances of winning her next games.
She estimates that if she has just won a game, she wins the next one in $70\%$ of cases. But if she has just suffered a defeat, according to her, the probability that she wins the next one is 0.2. Furthermore, she thinks she has an equal chance of winning the first game as of losing it.
For all non-zero natural integer $n$, we define the following events:

• $G _ { n }$: ``Léa wins the $n$-th game of the day'';
• $D _ { n }$: ``Léa loses the $n$-th game of the day''.

For all non-zero natural integer $n$, we denote $g _ { n }$ the probability of event $G _ { n }$. We have therefore $g _ { 1 } = 0.5$.

1. What is the value of the conditional probability $p _ { G _ { 1 } } \left( D _ { 2 } \right)$?
2. Copy and complete the probability tree below which models the situation for the first two games of the day.
3. Calculate $g _ { 2 }$.
4. Let $n$ be a non-zero natural integer. a. Copy and complete the probability tree below which models the situation for the $n$-th and $(n+1)$-th games of the day. b. Justify that for all non-zero natural integer $n$, $$g _ { n + 1 } = 0.5 g _ { n } + 0.2 .$$
5. For all non-zero natural integer $n$, we set $v _ { n } = g _ { n } - 0.4$. a. Show that the sequence $( v _ { n } )$ is geometric. We will specify its first term and its common ratio. b. Show that, for all non-zero natural integer $n$: $$g _ { n } = 0.1 \times 0.5 ^ { n - 1 } + 0.4 .$$
6. Study the variations of the sequence $( g _ { n } )$.
7. Give, by justifying, the limit of the sequence $( g _ { n } )$. Interpret the result in the context of the problem.
8. Determine, by calculation, the smallest integer $n$ such that $g _ { n } - 0.4 \leqslant 0.001$.
9. Copy and complete lines 4, 5 and 6 of the following function, written in Python language, so that it returns the smallest rank from which the terms of the sequence $\left( g _ { n } \right)$ are all less than or equal to $0.4 + e$, where $e$ is a strictly positive real number. \begin{verbatim} def seuil(e) : g = 0.5 n = 1 while...: g = 0.5 * g + 0.2 n = ... return (n) \end{verbatim}

A robot is positioned on a horizontal axis and moves several times by one meter on this axis, randomly to the right or to the left. During the first movement, the probability that the robot moves to the right is equal to $\frac{1}{3}$. If it moves to the right, the probability that the robot moves to the right again during the next movement is equal to $\frac{3}{4}$. If it moves to the left, the probability that the robot moves to the left again during the next movement is equal to $\frac{1}{2}$. For every natural integer $n \geqslant 1$, we denote:

• $D_n$ the event: ``the robot moves to the right during the $n$-th movement'';
• $\overline{D_n}$ the complementary event of $D_n$;
• $p_n$ the probability of event $D_n$.

We therefore have $p_1 = \frac{1}{3}$.
Part A: study of the special case where $n = 2$ In this part, the robot performs two successive movements.

1. Reproduce and complete the following weighted tree.
2. Determine the probability that the robot moves to the right twice.
3. Show that $p_2 = \frac{7}{12}$.
4. The robot moved to the left during the second movement. What is the probability that it moved to the right during the first movement?

Part B: study of the sequence $(p_n)$. We wish to estimate the movement of the robot after a large number of steps.

1. Prove that for every natural integer $n \geqslant 1$, we have: $$p_{n+1} = \frac{1}{4} p_n + \frac{1}{2}.$$ You may use a tree to help.
2. a. Show by induction that for every natural integer $n \geqslant 1$, we have: $$p_n \leqslant p_{n+1} < \frac{2}{3}.$$ b. Is the sequence $(p_n)$ convergent? Justify.
3. We consider the sequence $(u_n)$ defined for every natural integer $n \geqslant 1$, by $u_n = p_n - \frac{2}{3}$. a. Show that the sequence $(u_n)$ is geometric and specify its first term and its common ratio. b. Determine the limit of the sequence $(p_n)$ and interpret the result in the context of the exercise.

Part C In this part, we consider another robot that performs ten movements of one meter independent of each other, each movement to the right having a fixed probability equal to $\frac{3}{4}$. What is the probability that it returns to its starting point after the ten movements? Round the result to $10^{-3}$ near.

Exercise 1

We propose to compare the evolution of an animal population in two distinct environments A and B.
On January $1^{\text{st}}$ 2025, 6000 individuals are introduced into each of environments A and B.

Part A

In this part, we study the evolution of the population in environment A. We assume that in this environment, the evolution of the population is modelled by a geometric sequence $(u_n)$ with first term $u_0 = 6$ and common ratio 0.93. For every natural number $n$, $u_n$ represents the population on January $1^{\text{st}}$ of the year $2025 + n$, expressed in thousands of individuals.

1. Give, according to this model, the population on January $1^{\text{st}}$ 2026.
2. For every natural number $n$, express $u_n$ as a function of $n$.
3. Determine the limit of the sequence $(u_n)$.

Interpret this result in the context of the exercise.

Part B

In this part, we study the evolution of the population in environment B. We assume that in this environment, the evolution of the population is modelled by the sequence $(v_n)$ defined by
$$v_0 = 6 \text{ and for every natural number } n, v_{n+1} = -0.05 v_n^2 + 1.1 v_n.$$
For every natural number $n$, $v_n$ represents the population on January $1^{\text{st}}$ of the year $2025 + n$, expressed in thousands of individuals.

1. Give, according to this model, the population on January $1^{\text{st}}$ 2026.

Let $f$ be the function defined on the interval $[0; +\infty[$ by
$$f(x) = -0.05x^2 + 1.1x$$

1. Prove that the function $f$ is increasing on the interval $[0; 11]$.
2. Prove by induction that for every natural number $n$, we have $$2 \leqslant v_{n+1} \leqslant v_n \leqslant 6$$
3. Deduce that the sequence $(v_n)$ is convergent to a limit $\ell$.
4. a. Justify that the limit $\ell$ satisfies $f(\ell) = \ell$ then deduce the value of $\ell$. b. Interpret this result in the context of the exercise.

Part C

This part aims to compare the evolution of the population in the two environments.

1. By solving an inequality, determine the year from which the population of environment A will be strictly less than 3000 individuals.
2. Using a calculator, determine the year from which the population of environment B will be strictly less than 3000 individuals.
3. Justify that from a certain year onwards, the population of environment B will exceed the population of environment A.
4. Consider the Python program opposite. a. Copy and complete this program so that after execution, it displays the year from which the population of environment B is strictly greater than the population of environment A. b. Determine the year displayed after execution of the programme.

\begin{verbatim} n=0 u=6 v = 6 while...: u = ... v=... n = n+1 print (2025 + n) \end{verbatim}

Part B - Second model
After studying a larger collection of data over the last 50 years, another modelling appears more relevant:

• if the El Niño phenomenon is dominant in one year, then the probability that it is still dominant the following year is 0.5
• on the other hand, if the El Niño phenomenon is not dominant in one year, then the probability that it is dominant the following year is 0.3.

We consider that the reference year is 2023. We denote for every natural integer $n$:

• $E _ { n }$ the event ``the El Niño phenomenon is dominant in the year $2023 + n$ '';
• $p _ { n }$ the probability of the event $E _ { n }$.

In 2023, El Niño was not dominant. We thus have $p _ { 0 } = 0$.

1. Let $n$ be a natural integer. Copy and complete the following weighted tree.
2. Justify that $p _ { 1 } = 0.3$.
3. Using the tree, show that, for every natural integer $n$, we have: $$p _ { n + 1 } = 0.2 p _ { n } + 0.3$$
4. a. Conjecture the variations and the possible limit of the sequence $( p _ { n } )$. b. Show by induction that, for every natural integer $n$, we have: $p _ { n } \leqslant \frac { 3 } { 8 }$. c. Determine the direction of variation of the sequence $\left( p _ { n } \right)$. d. Deduce the convergence of the sequence $\left( p _ { n } \right)$.
5. Let $\left( u _ { n } \right)$ be the sequence defined by $u _ { n } = p _ { n } - \frac { 3 } { 8 }$ for every natural integer $n$. a. Show that the sequence $( u _ { n } )$ is geometric with ratio 0.2 and specify its first term. b. Show that, for every natural integer $n$, we have: $$p _ { n } = \frac { 3 } { 8 } \left( 1 - 0.2 ^ { n } \right) .$$ c. Calculate the limit of the sequence $\left( p _ { n } \right)$. d. Interpret this result in the context of the exercise.

Let $n$ be a non-zero natural integer. In the context of a random experiment, we consider a sequence of events $A _ { n }$ and we denote $p _ { n }$ the probability of the event $A _ { n }$. For parts $\mathbf { A }$ and $\mathbf { B }$ of the exercise, we consider that:

• If the event $A _ { n }$ is realized then the event $A _ { n + 1 }$ is realized with probability 0.3.
• If the event $A _ { n }$ is not realized then the event $A _ { n + 1 }$ is realized with probability 0.7.

We assume that $p _ { 1 } = 1$.

Part A:

1. Copy and complete the probabilities on the branches of the probability tree below.
2. Show that $p _ { 3 } = 0.58$.
3. Calculate the conditional probability $P _ { A _ { 3 } } \left( A _ { 2 } \right)$, round the result to $10 ^ { - 2 }$ near.

Part B:

In this part, we study the sequence $( p _ { n } )$ with $n \geqslant 1$.

1. Copy and complete the probabilities on the branches of the probability tree below.
2. a. Show that, for all non-zero natural integer $n$: $p _ { n + 1 } = - 0.4 p _ { n } + 0.7$.

We consider the sequence $( u _ { n } )$, defined for all non-zero natural integer $n$ by: $u _ { n } = p _ { n } - 0.5$. b. Show that $( u _ { n } )$ is a geometric sequence for which we will specify the common ratio and the first term. c. Deduce the expression of $u _ { n }$, then of $p _ { n }$ as a function of $n$. d. Determine the limit of the sequence $\left( p _ { n } \right)$.

Part C:

Let $x \in ] 0 ; 1 [$, we assume that $P _ { \overline { A _ { n } } } \left( A _ { n + 1 } \right) = P _ { A _ { n } } \left( \overline { A _ { n + 1 } } \right) = x$. We recall that $p _ { 1 } = 1$.

1. Show that for all non-zero natural integer $n$: $p _ { n + 1 } = ( 1 - 2 x ) p _ { n } + x$.
2. Prove by induction on $n$ that, for all non-zero natural integer $n$: $$p _ { n } = \frac { 1 } { 2 } ( 1 - 2 x ) ^ { n - 1 } + \frac { 1 } { 2 }$$
3. Show that the sequence $\left( p _ { n } \right)$ is convergent and give its limit.

Question 167
A soma dos termos de uma progressão geométrica finita de razão $q = 2$, primeiro termo $a_1 = 1$ e $n = 5$ termos é
(A) 15 (B) 20 (C) 31 (D) 32 (E) 63

Question 176
Um banco oferece uma aplicação com juros compostos de 1\% ao mês. Um cliente aplica R\$ 10 000,00. Após 2 meses, o montante obtido será de
(A) R\$ 10 100,00 (B) R\$ 10 200,00 (C) R\$ 10 201,00 (D) R\$ 10 210,00 (E) R\$ 10 220,00

Uma progressão geométrica tem primeiro termo $a_1 = 2$ e razão $q = 3$. A soma dos quatro primeiros termos dessa progressão é
(A) 26 (B) 40 (C) 54 (D) 80 (E) 162

A soma infinita da progressão geométrica com primeiro termo $a_1 = 6$ e razão $q = \dfrac{1}{3}$ é
(A) 6 (B) 7 (C) 8 (D) 9 (E) 10

QUESTION 147
In a geometric progression, the first term is 2 and the common ratio is 3. The fifth term of this progression is
(A) 54
(B) 81
(C) 162
(D) 243
(E) 486

To celebrate a city's anniversary, the city council organizes four consecutive days of cultural attractions. Experience from previous years shows that, from one day to the next, the number of visitors to the event is tripled. 345 visitors are expected for the first day of the event.
A possible representation of the expected number of participants for the last day is
(A) $3 \times 345$
(B) $(3 + 3 + 3) \times 345$
(C) $3^{3} \times 345$
(D) $3 \times 4 \times 345$
(E) $3^{4} \times 345$

A geometric progression has first term 2 and common ratio 3. What is the sum of the first 4 terms?
(A) 60
(B) 70
(C) 80
(D) 90
(E) 100

(i) By the binomial theorem $(\sqrt{2} + 1)^{10} = \sum_{i=0}^{10} C_i (\sqrt{2})^i$, where $C_i$ are appropriate constants. Write the value of $i$ for which $C_i (\sqrt{2})^i$ is the largest among the 11 terms in this sum.
(ii) For every natural number $n$, let $(\sqrt{2} + 1)^n = p_n + \sqrt{2} q_n$, where $p_n$ and $q_n$ are integers. Calculate $\lim_{n \rightarrow \infty} \left(\frac{p_n}{q_n}\right)^{10}$.

Consider an equilateral triangle $ABC$ with altitude 3 centimeters. A circle is inscribed in this triangle, then another circle is drawn such that it is tangent to the inscribed circle and the sides $AB$, $AC$. Infinitely many such circles are drawn; each tangent to the previous circle and the sides $AB$, $AC$. Find the sum of the areas of all these circles.

Suppose $a_0, a_1, a_2, a_3, \ldots$ is an arithmetic progression with $a_0$ and $a_1$ positive integers. Let $g_0, g_1, g_2, g_3, \ldots$ be the geometric progression such that $g_0 = a_0$ and $g_1 = a_1$.
Statements
(1) We must have $\left(a_5\right)^2 \geq a_0 a_{10}$.
(2) The sum $a_0 + a_1 + \cdots + a_{10}$ must be a multiple of the integer $a_5$.
(3) If $\sum_{i=0}^{\infty} a_i$ is $+\infty$ then $\sum_{i=0}^{\infty} g_i$ is also $+\infty$.
(4) If $\sum_{i=0}^{\infty} g_i$ is finite then $\sum_{i=0}^{\infty} a_i$ is $-\infty$.

A society where the proportion of the population aged 65 and over in the total population is 20\% or more is called a 'super-aged society'. In 2000, a certain country had a total population of 10 million and a population aged 65 and over of 500,000. Assuming that the total population increases by 0.3\% each year compared to the previous year and the population aged 65 and over increases by 4\% each year compared to the previous year, when is the first time a 'super-aged society' predicted to occur? (Given: $\log 1.003 = 0.0013$, $\log 1.04 = 0.0170$, $\log 2 = 0.3010$) [4 points]
(1) 2048--2050
(2) 2038--2040
(3) 2028--2030
(4) 2018--2020
(5) 2008--2010

A society where the proportion of the population aged 65 and over is 20\% or more of the total population is called a 'super-aged society'.
In the year 2000, a certain country had a total population of 10 million and a population aged 65 and over of 500,000. Assuming that the total population increases by 0.3\% each year compared to the previous year and the population aged 65 and over increases by 4\% each year compared to the previous year, when is the first time a 'super-aged society' predicted to occur? (Given: $\log 1.003 = 0.0013 , \log 1.04 = 0.0170 , \log 2 = 0.3010$) [4 points]
(1) 2048--2050
(2) 2038--2040
(3) 2028--2030
(4) 2018--2020
(5) 2008--2010

For a geometric sequence $\left\{ a _ { n } \right\}$ with common ratio $r$ and $a _ { 2 } = 1$, let $\omega = a _ { 1 } a _ { 2 } a _ { 3 } \cdots a _ { 10 }$ be the product of the first 10 terms. Find the value of $\log _ { r } \omega$. (Here, $r > 0$ and $r \neq 1$.) [3 points]

As shown in the figure below, from a square with side length 1, a square with side length $\frac { 1 } { 2 }$ is cut out, and the remaining shape is called $A _ { 1 }$. From a square with side length $\frac { 1 } { 4 }$, a square with side length $\frac { 1 } { 8 }$ is cut out, and two resulting shapes are attached to the upper two sides of $A _ { 1 }$ to form a figure called $A _ { 2 }$. From a square with side length $\frac { 1 } { 16 }$, a square with side length $\frac { 1 } { 32 }$ is cut out, and four resulting shapes are attached to the upper four sides of $A _ { 2 }$ to form a figure called $A _ { 3 }$.
Continuing this process, let $A _ { n }$ be the $n$-th figure obtained and $S _ { n }$ be its area. If $\lim _ { n \rightarrow \infty } S _ { n } = \frac { q } { p }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

For an infinite geometric sequence $\left\{ a _ { n } \right\}$, choose all correct statements from \textless Remarks\textgreater. [3 points]
\textless Remarks\textgreater ㄱ. If the infinite geometric series $\sum _ { n = 1 } ^ { \infty } a _ { n }$ converges, then $\sum _ { n = 1 } ^ { \infty } a _ { 2 n }$ also converges. ㄴ. If the infinite geometric series $\sum _ { n = 1 } ^ { \infty } a _ { n }$ diverges, then $\sum _ { n = 1 } ^ { \infty } a _ { 2 n }$ also diverges. ㄷ. If the infinite geometric series $\sum _ { n = 1 } ^ { \infty } a _ { n }$ converges, then $\sum _ { n = 1 } ^ { \infty } \left( a _ { n } + \frac { 1 } { 2 } \right)$ also converges.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄱ, ㄴ
(5) ㄴ, ㄷ

There is a right isosceles triangle with the two legs forming the right angle each having length 1. A square is drawn such that 2 of its vertices lie on the hypotenuse and the other 2 vertices lie on the two legs forming the right angle. The figure obtained by coloring this square is called $R _ { 1 }$. In figure $R _ { 1 }$, 2 congruent right isosceles triangles are formed. In each of these triangles, a square is drawn such that 2 of its vertices lie on the hypotenuse and the other 2 vertices lie on the two legs forming the right angle. The figure obtained by coloring these 2 squares is called $R _ { 2 }$. In figure $R _ { 2 }$, 4 congruent right isosceles triangles are formed. In each of these triangles, a square is drawn such that 2 of its vertices lie on the hypotenuse and the other 2 vertices lie on the two legs forming the right angle. The figure obtained by coloring these 4 squares is called $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the sum of the areas of all colored squares in the $n$-th figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 3 \sqrt { 2 } } { 20 }$
(2) $\frac { \sqrt { 2 } } { 5 }$
(3) $\frac { 3 } { 10 }$
(4) $\frac { \sqrt { 3 } } { 5 }$
(5) $\frac { 2 } { 5 }$

For a geometric sequence $\left\{ a _ { n } \right\}$, when $a _ { 3 } = 2 , a _ { 6 } = 16$, find the value of $a _ { 9 }$. [3 points]

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]

Inside a rectangle with width 6 and height 8, a circle with center at the intersection of the two diagonals and diameter equal to $\frac { 1 } { 3 }$ of the width of the rectangle is drawn to obtain figure $R _ { 1 }$. From figure $R _ { 1 }$, four rectangles are drawn with each segment from a vertex of the rectangle to the intersection of the diagonal and circle as the diagonal. Inside each of the newly drawn rectangles, a circle with center at the intersection of the two diagonals and diameter equal to $\frac { 1 } { 3 }$ of the width of the newly drawn rectangle is drawn to obtain figure $R _ { 2 }$.
In figure $R _ { 2 }$, for each of the four congruent rectangles, four rectangles are drawn with each segment from a vertex to the intersection of the diagonal and circle as the diagonal. Inside each of the newly drawn rectangles, a circle with center at the intersection of the two diagonals and diameter equal to $\frac { 1 } { 3 }$ of the width of the newly drawn rectangle is drawn to obtain figure $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the sum of the areas of all circles in the $n$-th obtained figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? (Here, the widths and heights of all rectangles are parallel to each other, respectively.) [4 points]
(1) $\frac { 37 } { 9 } \pi$
(2) $\frac { 34 } { 9 } \pi$
(3) $\frac { 31 } { 9 } \pi$
(4) $\frac { 28 } { 9 } \pi$
(5) $\frac { 25 } { 9 } \pi$