In a factory, a kiln bakes ceramics at a temperature of $1000^{\circ}\mathrm{C}$. At the end of baking, it is turned off and cools down. The temperature of the kiln is expressed in degrees Celsius (${}^{\circ}\mathrm{C}$). The kiln door can be opened safely for the ceramics as soon as its temperature is below $70^{\circ}\mathrm{C}$.
For a natural integer $n$, we denote $T_n$ the temperature in degrees Celsius of the kiln after $n$ hours have elapsed from the moment it was turned off. We therefore have $T_0 = 1000$. The temperature $T_n$ is calculated by the following algorithm:
\begin{verbatim} T←1000 For i going from 1 to n T←0.82 x T+3.6 End For \end{verbatim}

1. Determine the temperature of the kiln, rounded to the nearest unit, after 4 hours of cooling.
2. Prove that, for every natural integer $n$, we have: $T_n = 980 \times 0.82^n + 20$.
3. After how many hours can the kiln be opened safely for the ceramics?

A patient must take a dose of 2 mL of a medication every hour.\nWe introduce the sequence $\left( u _ { n } \right)$ such that the term $u _ { n }$ represents the quantity of medication, expressed in mL, present in the body immediately after $n$ doses of medication.\nWe have $u _ { 1 } = 2$ and for every strictly positive natural number $n$: $u _ { n + 1 } = 2 + 0.8 u _ { n }$.

Part A

Using this model, a doctor seeks to determine after how many doses of medication the quantity present in the patient's body is strictly greater than 9 mL.

1. Calculate the value $u _ { 2 }$.
2. Show, by induction on $n$, that $u _ { n } = 10 - 8 \times 0.8 ^ { n - 1 }$ for every strictly positive natural number $n$.
3. Determine $\lim _ { n \rightarrow + \infty } u _ { n }$ and give an interpretation of this result in the context of the exercise.
4. Let $N$ be a strictly positive natural number, does the inequality $u _ { N } \geq 10$ have solutions? Interpret the result of this question in the context of the exercise.
5. Determine from how many doses of medication the quantity of medication present in the patient's body is strictly greater than 9 mL. Justify your approach.

Part B

Using the same modeling, the doctor is interested in the average quantity of medication present in the patient's body over time.\nFor this purpose, we define the sequence ( $S _ { n }$ ) defined for every strictly positive natural number $n$ by
$$S _ { n } = \frac { u _ { 1 } + u _ { 2 } + \cdots + u _ { n } } { n }$$
We admit that the sequence ( $S _ { n }$ ) is increasing.

1. Calculate $S _ { 2 }$.
2. Show that, for every strictly positive natural number $n$,

$$u _ { 1 } + u _ { 2 } + \cdots + u _ { n } = 10 n - 40 + 40 \times 0.8 ^ { n }$$

1. Calculate $\lim _ { n \rightarrow + \infty } S _ { n }$.
2. The following mystery function is given, written in Python language.

\begin{verbatim} def mystere(k): $\mathrm { n } = 1$ $\mathrm { s } = 2$ while $\mathrm { s } < \mathrm { k }$ : $\mathrm { n } = \mathrm { n } + 1$ $\mathrm { s } = 10 - 40 / \mathrm { n } + ( 40 * 0.8 * * \mathrm { n } ) / \mathrm { n }$ return $n$ \end{verbatim}
In the context of the statement, what does the value returned by the input mystere (9) represent?\nDetermine the value returned by the input mystere (9).

1. Justify that this value is strictly greater than 10.

In the sequence $\left\{ a _ { n } \right\}$, $a _ { 1 } = 1 , a _ { 2 } = 4 , a _ { 3 } = 10$, and the sequence $\left\{ a _ { n + 1 } - a _ { n } \right\}$ is a geometric sequence. Find the value of $a _ { 5 }$. [3 points]

A sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { n + 1 } - a _ { n } = 2 n$. When $a _ { 10 } = 94$, what is the value of $a _ { 1 }$? [3 points]
(1) 5
(2) 4
(3) 3
(4) 2
(5) 1

For a sequence $\left\{ a _ { n } \right\}$, let $S _ { n }$ denote the sum of the first $n$ terms. The sequence $\left\{ S _ { 2 n - 1 } \right\}$ is an arithmetic sequence with common difference $-3$, and the sequence $\left\{ S _ { 2 n } \right\}$ is an arithmetic sequence with common difference $2$. When $a _ { 2 } = 1$, find the value of $a _ { 8 }$. [4 points]

For a sequence $\left\{ a _ { n } \right\}$ with $a _ { 1 } = 1$ and satisfying
$$a _ { n + 1 } = \frac { 2 n } { n + 1 } a _ { n }$$
for all natural numbers $n$, what is the value of $a _ { 4 }$? [3 points]
(1) $\frac { 3 } { 2 }$
(2) 2
(3) $\frac { 5 } { 2 }$
(4) 3
(5) $\frac { 7 } { 2 }$

For natural numbers $n$, the point $\mathrm{P}_n$ on the coordinate plane is determined according to the following rules. (가) The coordinates of the three points $\mathrm{P}_1, \mathrm{P}_2, \mathrm{P}_3$ are $(-1, 0)$, $(1, 0)$, and $(-1, 2)$, respectively. (나) The midpoint of segment $\mathrm{P}_n \mathrm{P}_{n+1}$ and the midpoint of segment $\mathrm{P}_{n+2} \mathrm{P}_{n+3}$ are the same. For example, the coordinates of point $\mathrm{P}_4$ are $(1, -2)$. If the coordinates of point $\mathrm{P}_{25}$ are $(a, b)$, find the value of $a + b$. [4 points]

For a natural number $n$, the point $\mathrm { P } _ { n }$ on the coordinate plane is determined according to the following rules.
(a) The coordinates of the three points $\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 }$ are $( - 1,0 ) , ( 1,0 )$, and $( - 1,2 )$, respectively.
(b) The midpoint of line segment $\mathrm { P } _ { n } \mathrm { P } _ { n + 1 }$ and the midpoint of line segment $\mathrm { P } _ { n + 2 } \mathrm { P } _ { n + 3 }$ are the same. For example, the coordinates of point $\mathrm { P } _ { 4 }$ are $( 1 , - 2 )$. When the coordinates of point $\mathrm { P } _ { 25 }$ are $( a , b )$, find the value of $a + b$. [4 points]

A sequence $\left\{ a _ { n } \right\}$ satisfies the following conditions. (가) $a _ { 1 } = a _ { 2 } + 3$ (나) $a _ { n + 1 } = - 2 a _ { n } ( n \geq 1 )$ Find the value of $a _ { 9 }$. [3 points]

17. (12 points)
A sequence $\left\{ a _ { n } \right\}$ satisfies $\frac { 1 } { a _ { n + 1 } } - \frac { 2 } { a _ { n } } = 0$, and $a _ { 1 } = \frac { 1 } { 2 }$.
(1) Find the general term formula of the sequence $\left\{ a _ { n } \right\}$;
(2) Find the sum $S _ { n }$ of the first $n$ terms of the sequence $\left\{ \frac { 1 } { a _ { n } } + 2 n \right\}$.

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ be an arithmetic progression with $a _ { 1 } = 7$ and common difference 8. Let $T _ { 1 } , T _ { 2 } , T _ { 3 } , \ldots$ be such that $T _ { 1 } = 3$ and $T _ { n + 1 } - T _ { n } = a _ { n }$ for $n \geq 1$. Then, which of the following is/are TRUE ?
(A) $T _ { 20 } = 1604$
(B) $\sum _ { k = 1 } ^ { 20 } T _ { k } = 10510$
(C) $T _ { 30 } = 3454$
(D) $\sum _ { k = 1 } ^ { 30 } T _ { k } = 35610$

The sequences $\{a_{n}\}$ and $\{b_{n}\}$ are defined as follows. $$a_{n} = \begin{cases} 0, & \text{if } n \equiv 0 \pmod{3} \\ n, & \text{if } n \equiv 1 \pmod{3} \\ -n, & \text{if } n \equiv 2 \pmod{3} \end{cases}, \quad b_{n} = \sum_{k=0}^{n} a_{k}$$ Accordingly, what is $b_{4}$?
A) $-2$
B) $-1$
C) $0$
D) $2$
E) $3$