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?

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) $3$