We are given the sequence $( u _ { n } )$ defined by: $u _ { 0 } = 0$ and for every natural integer $n$, $u _ { n + 1 } = \frac { 1 } { 2 } u _ { n } + 1$. The sequence $\left( v _ { n } \right)$, defined for every natural integer $n$ by $v _ { n } = u _ { n } - 2$, is: a. arithmetic with common difference $- 2$; b. geometric with common ratio $- 2$; c. arithmetic with common difference $1$; d. geometric with common ratio $\frac { 1 } { 2 }$.

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 A

Let $\left(u_{n}\right)$ be the sequence defined by $u_{0} = 30$ and, for every natural integer $n$, $u_{n+1} = \frac{1}{2} u_{n} + 10$. Let $(v_{n})$ be the sequence defined for every natural integer $n$ by $v_{n} = u_{n} - 20$.

1. Calculate the exact values of $u_{1}$ and $u_{2}$.
2. Prove that the sequence $(v_{n})$ is geometric with common ratio $\frac{1}{2}$.
3. Express $v_{n}$ as a function of $n$ for every natural integer $n$.
4. Deduce that, for every natural integer $n$, $u_{n} = 20 + 10\left(\frac{1}{2}\right)^{n}$.
5. Determine the limit of the sequence $\left(u_{n}\right)$. Justify the answer.

Part B

Let $(w_{n})$ be the sequence defined for every natural integer $n$ by: $$\left\{\begin{array}{l} w_{0} = 45 \\ w_{n+1} = \frac{1}{2} w_{n} + \frac{1}{2} u_{n} + 7 \end{array}\right.$$

1. Show that $w_{1} = 44.5$.

We wish to write a function \texttt{suite}, in Python language, which returns the value of the term $w_{n}$ for a given value of $n$. We give below a proposal for this function \texttt{suite}. \begin{verbatim} def suite(n) : U=30 W=45 for i in range (1,n+1) : U=U/2+10 W=W/2+U/2+7 return W \end{verbatim}

1. The execution of \texttt{suite(1)} does not return the term $w_{1}$. How should the function \texttt{suite} be modified so that the execution of \texttt{suite(n)} returns the value of the term $w_{n}$?
2. a. Show, by induction on $n$, that for every natural integer $n$ we have: $$w_{n} = 10n\left(\frac{1}{2}\right)^{n} + 11\left(\frac{1}{2}\right)^{n} + 34$$ b. We admit that for every natural integer $n \geqslant 4$, we have: $0 \leqslant 10n\left(\frac{1}{2}\right)^{n} \leqslant \frac{10}{n}$. What can we deduce about the convergence of the sequence $\left(w_{n}\right)$?

Part A

Let $\left( u _ { n } \right)$ be the sequence defined by $u _ { 0 } = 30$ and, for every natural integer $n$, $u _ { n + 1 } = \frac { 1 } { 2 } u _ { n } + 10$. Let $\left( v _ { n } \right)$ be the sequence defined for every natural integer $n$ by $v _ { n } = u _ { n } - 20$.

1. Calculate the exact values of $u _ { 1 }$ and $u _ { 2 }$.
2. Prove that the sequence ( $v _ { n }$ ) is geometric with ratio $\frac { 1 } { 2 }$.
3. Express $v _ { n }$ as a function of $n$ for every natural integer $n$.
4. Deduce that, for every natural integer $n$, $u _ { n } = 20 + 10 \left( \frac { 1 } { 2 } \right) ^ { n }$.
5. Determine the limit of the sequence ( $u _ { n }$ ). Justify the answer.

Part B

Let ( $w _ { n }$ ) be the sequence defined for every natural integer $n$ by:
$$\left\{ \begin{array} { l } w _ { 0 } = 45 \\ w _ { n + 1 } = \frac { 1 } { 2 } w _ { n } + \frac { 1 } { 2 } u _ { n } + 7 \end{array} \right.$$

1. Show that $w _ { 1 } = 44.5$.

We wish to write a function suite, in Python language, which returns the value of the term $w _ { n }$ for a given value of $n$. We give below a proposal for this function suite.
\begin{verbatim} def suite(n): U=30 W=45 for i in range (1,n+1): \mathrm { U } = \mathrm { U } / 2 + 1 0 W=W/2+U/2+7 return W \end{verbatim}

1. The execution of suite(1) does not return the term $w _ { 1 }$. How should the function suite be modified so that the execution of suite( $n$ ) returns the value of the term $w _ { n }$?
2. (a) Show, by induction on $n$, that for every natural integer $n$ we have:

$$w _ { n } = 10 n \left( \frac { 1 } { 2 } \right) ^ { n } + 11 \left( \frac { 1 } { 2 } \right) ^ { n } + 34$$
(b) It is admitted that for every natural integer $n \geq 4$, we have: $0 \leq 10 n \left( \frac { 1 } { 2 } \right) ^ { n } \leq \frac { 10 } { n }$.
What can we deduce about the convergence of the sequence $\left( w _ { n } \right)$?

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

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

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.

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

The following is a probability distribution table for the random variable $X$.

$X$	$k$	$2 k$	$4 k$	Total
$\mathrm { P } ( X = x )$	$\frac { 4 } { 7 }$	$a$	$b$	1

If $\frac { 4 } { 7 } , a , b$ form a geometric sequence in this order and the mean of $X$ is 24, find the value of $k$. [3 points]

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

Four numbers $1 , a , b , c$ form a geometric sequence with common ratio $r$ in this order, and satisfy $\log _ { 8 } c = \log _ { a } b$. What is the value of the common ratio $r$? (where $r > 1$) [3 points]
(1) 2
(2) $\frac { 5 } { 2 }$
(3) 3
(4) $\frac { 7 } { 2 }$
(5) 4

For two infinite geometric sequences $\left\{ a _ { n } \right\} , \left\{ b _ { n } \right\}$ with the same common ratio, $a _ { 1 } - b _ { 1 } = 1$, $\sum _ { n = 1 } ^ { \infty } a _ { n } = 8$, and $\sum _ { n = 1 } ^ { \infty } b _ { n } = 6$. Find the value of $\sum _ { n = 1 } ^ { \infty } a _ { n } b _ { n }$. [3 points]

A geometric sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 2 } = \frac { 1 } { 2 }$ and $a _ { 5 } = \frac { 1 } { 6 }$. When $\sum _ { n = 1 } ^ { \infty } a _ { n } a _ { n + 1 } a _ { n + 2 } = \frac { q } { p }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

For two natural numbers $a$ and $b$, the three numbers $a ^ { n } , 2 ^ { 4 } \times 3 ^ { 6 } , b ^ { n }$ form a geometric sequence in this order. Find the minimum value of $a b$. (Here, $n$ is a natural number.) [4 points]

On the coordinate plane, for a natural number $n$, the coordinates of point $\mathrm { P } _ { n }$ are $\left( n , 3 ^ { n } \right)$ and the coordinates of point $\mathrm { Q } _ { n }$ are $( n , 0 )$. Let $a _ { n }$ be the area of the quadrilateral $\mathrm { P } _ { n } \mathrm { Q } _ { n + 1 } \mathrm { Q } _ { n + 2 } \mathrm { P } _ { n + 1 }$. When $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { a _ { n } } = \frac { q } { p }$, find the value of $p ^ { 2 } + q ^ { 2 }$. (where $p$ and $q$ are coprime natural numbers) [4 points]

For a geometric sequence $\left\{ a _ { n } \right\}$ with all positive terms, $$\frac { a _ { 1 } a _ { 2 } } { a _ { 3 } } = 2 , \quad \frac { 2a _ { 2 } } { a _ { 1 } } + \frac { a _ { 4 } } { a _ { 2 } } = 8$$ what is the value of $a _ { 3 }$? [3 points]
(1) 16
(2) 18
(3) 20
(4) 22
(5) 24

For a geometric sequence $\left\{ a _ { n } \right\}$ with positive common ratio, if $a _ { 1 } = 3 , a _ { 5 } = 48$, what is the value of $a _ { 3 }$? [3 points]
(1) 18
(2) 16
(3) 14
(4) 12
(5) 10

For a geometric sequence $\left\{ a _ { n } \right\}$ with $a _ { 1 } = 3 , a _ { 2 } = 1$, what is the value of $\sum _ { n = 1 } ^ { \infty } \left( a _ { n } \right) ^ { 2 }$? [3 points]
(1) $\frac { 89 } { 8 }$
(2) $\frac { 87 } { 8 }$
(3) $\frac { 85 } { 8 }$
(4) $\frac { 83 } { 8 }$
(5) $\frac { 81 } { 8 }$