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

The sequence $\left\{ a _ { n } \right\}$ has $a _ { 1 } = 2$, and for all natural numbers $n$, $$a _ { n + 1 } = \begin{cases} a _ { n } - 1 & \text{(when } a _ { n } \text{ is even)} \\ a _ { n } + n & \text{(when } a _ { n } \text{ is odd)} \end{cases}$$ Find the value of $a _ { 7 }$. [3 points]
(1) 7
(2) 9
(3) 11
(4) 13
(5) 15

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

A sequence $\left\{ a _ { n } \right\}$ has $a _ { 1 } = 2$ and satisfies for all natural numbers $n$: $$a _ { n + 1 } = \left\{ \begin{array} { c c } \frac { a _ { n } } { 2 - 3 a _ { n } } & ( \text{when } n \text{ is odd} ) \\ 1 + a _ { n } & ( \text{when } n \text{ is even} ) \end{array} \right.$$ What is the value of $\sum _ { n = 1 } ^ { 40 } a _ { n }$? [3 points]
(1) 30
(2) 35
(3) 40
(4) 45
(5) 50

What is the value of $\lim _ { n \rightarrow \infty } \frac { \sqrt { 9 n ^ { 2 } + 4 } } { 5 n - 2 }$? [2 points]
(1) $\frac { 1 } { 5 }$
(2) $\frac { 2 } { 5 }$
(3) $\frac { 3 } { 5 }$
(4) $\frac { 4 } { 5 }$
(5) 1

The sequence $\left\{ a _ { n } \right\}$ satisfies the following conditions for all natural numbers $n$. (가) $a _ { 2 n } = a _ { n } - 1$ (나) $a _ { 2 n + 1 } = 2 a _ { n } + 1$ When $a _ { 20 } = 1$, what is the value of $\sum _ { n = 1 } ^ { 63 } a _ { n }$? [4 points]
(1) 704
(2) 712
(3) 720
(4) 728
(5) 736

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

A sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } = 1$ and for all natural numbers $n$, $$\sum _ { k = 1 } ^ { n } \left( a _ { k } - a _ { k + 1 } \right) = - n ^ { 2 } + n$$ What is the value of $a _ { 11 }$? [3 points]
(1) 88
(2) 91
(3) 94
(4) 97
(5) 100

A sequence $\left\{ a _ { n } \right\}$ satisfies $0 < a _ { 1 } < 1$ and the following conditions for all natural numbers $n$: (가) $a _ { 2 n } = a _ { 2 } \times a _ { n } + 1$ (나) $a _ { 2 n + 1 } = a _ { 2 } \times a _ { n } - 2$ If $a _ { 7 } = 2$, what is the value of $a _ { 25 }$? [4 points]
(1) 78
(2) 80
(3) 82
(4) 84
(5) 86

The sequence $\left\{ a _ { n } \right\}$ satisfies $0 < a _ { 1 } < 1$ and the following conditions for all natural numbers $n$. (가) $a _ { 2 n } = a _ { 2 } \times a _ { n } + 1$ (나) $a _ { 2 n + 1 } = a _ { 2 } \times a _ { n } - 2$ When $a _ { 8 } - a _ { 15 } = 63$, what is the value of $\frac { a _ { 8 } } { a _ { 1 } }$? [4 points]
(1) 91
(2) 92
(3) 93
(4) 94
(5) 95

For a sequence $\left\{ a _ { n } \right\}$ with first term 1, satisfying for all natural numbers $n$: $$a _ { n + 1 } = \begin{cases} 2 a _ { n } & \left( a _ { n } < 7 \right) \\ a _ { n } - 7 & \left( a _ { n } \geq 7 \right) \end{cases}$$ what is the value of $\sum _ { k = 1 } ^ { 8 } a _ { k }$? [3 points]
(1) 30
(2) 32
(3) 34
(4) 36
(5) 38

For all sequences $\left\{ a _ { n } \right\}$ with all natural number terms satisfying the following conditions, let $M$ and $m$ be the maximum and minimum values of $a _ { 9 }$, respectively. What is the value of $M + m$? [4 points] (가) $a _ { 7 } = 40$ (나) For all natural numbers $n$, $$a _ { n + 2 } = \begin{cases} a _ { n + 1 } + a _ { n } & ( \text{when } a _ { n + 1 } \text{ is not a multiple of } 3 ) \\ \frac { 1 } { 3 } a _ { n + 1 } & ( \text{when } a _ { n + 1 } \text{ is a multiple of } 3 ) \end{cases}$$ (1) 216
(2) 218
(3) 220
(4) 222
(5) 224

A sequence $\{a_n\}$ with a natural number as its first term satisfies $$a_{n+1} = \begin{cases} 2^{a_n} & (\text{if } a_n \text{ is odd}) \\ \frac{1}{2}a_n & (\text{if } a_n \text{ is even}) \end{cases}$$ for all natural numbers $n$. Find the sum of all values of $a_1$ such that $a_6 + a_7 = 3$. [4 points]
(1) 139
(2) 146
(3) 153
(4) 160
(5) 167

For a sequence $\left\{ a_{n} \right\}$, $\lim_{n \rightarrow \infty} \frac{n a_{n}}{n^{2} + 3} = 1$. What is the value of $\lim_{n \rightarrow \infty} \left(\sqrt{a_{n}^{2} + n} - a_{n}\right)$? [3 points]
(1) $\frac{1}{3}$
(2) $\frac{1}{2}$
(3) $1$
(4) $2$
(5) $3$

The sequence $\left\{ a _ { n } \right\}$ has $a _ { 1 } = 1$ and satisfies $$a _ { n + 1 } = n ^ { 2 } a _ { n } + 1$$ for all natural numbers $n$. Find the value of $a _ { 3 }$. [3 points]

The sequence $\left\{ a _ { n } \right\}$ satisfies the following conditions.

• $a _ { 1 } = 7$
• For natural numbers $n \geq 2$,

$$\sum _ { k = 1 } ^ { n } a _ { k } = \frac { 2 } { 3 } a _ { n } + \frac { 1 } { 6 } n ^ { 2 } - \frac { 1 } { 6 } n + 10$$
The following is the process of finding the value of $\sum _ { k = 1 } ^ { 12 } a _ { k } + \sum _ { k = 1 } ^ { 5 } a _ { 2 k + 1 }$.
For natural numbers $n \geq 2$, since $a _ { n + 1 } = \sum _ { k = 1 } ^ { n + 1 } a _ { k } - \sum _ { k = 1 } ^ { n } a _ { k }$, $$a _ { n + 1 } = \frac { 2 } { 3 } \left( a _ { n + 1 } - a _ { n } \right) + \text { (가) }$$ and simplifying this equation gives $$2 a _ { n } + a _ { n + 1 } = 3 \times \text { (가) } \quad \cdots \cdots \text { (ㄱ) }$$ From $$\sum _ { k = 1 } ^ { n } a _ { k } = \frac { 2 } { 3 } a _ { n } + \frac { 1 } { 6 } n ^ { 2 } - \frac { 1 } { 6 } n + 10 \quad ( n \geq 2 )$$ substituting $n = 2$ gives $$a _ { 2 } = \text { (나) }$$ By (ㄱ) and (ㄴ), $$\begin{aligned} \sum _ { k = 1 } ^ { 12 } a _ { k } + \sum _ { k = 1 } ^ { 5 } a _ { 2 k + 1 } & = a _ { 1 } + a _ { 2 } + \sum _ { k = 1 } ^ { 5 } \left( 2 a _ { 2 k + 1 } + a _ { 2 k + 2 } \right) \\ & = \text { (다) } \end{aligned}$$ Let $f ( n )$ be the expression that fits in (가), and let $p$ and $q$ be the numbers that fit in (나) and (다), respectively. Find the value of $\frac { p \times q } { f ( 12 ) }$. [4 points]

14. The three lines $l _ { 1 } : x + y - 1 = 0 , l _ { 2 } : n x + y - n = 0 , l _ { 3 } : x + n y - n = 0 \left( n \in \mathbf { N } ^ { * } , n \geq 2 \right)$ form a triangle with area denoted as $S _ { n }$ .
Then $\lim _ { n \rightarrow \infty } S _ { n } =$ $\_\_\_\_$.

II. Multiple Choice (Total Score: 20 points, 5 points each)

21. (Total Score: 14 points) Subproblem 1: 6 points, Subproblem 2: 8 points. Given that the sum of the first $n$ terms of sequence $\left\{ a _ { n } \right\}$ is $S _ { n }$ , and $S _ { n } = n - 5 a _ { n } - 85 , n \in \mathbf { N } ^ { * }$ .
(1) Prove that $\left\{ a _ { n } - 1 \right\}$ is a geometric sequence;
(2) Find the general term formula for the sequence $\left\{ S _ { n } \right\}$ , and find the minimum positive integer $n$ such that $S _ { n + 1 } > S _ { n }$ .

2. Calculate $\lim_{n \rightarrow \infty} \left( 1 - \frac{3n}{n+3} \right) =$ $\_\_\_\_$

Let $\mathrm { S } _ { \mathrm { n } }$ be the sum of the first $n$ terms of sequence $\left\{ \mathrm { a } _ { \mathrm { n } } \right\}$, and $a _ { 1 } = - 1 , a _ { \mathrm { n } + 1 } = S _ { n } S _ { n + 1 }$. Then $S _ { n } = $ $\_\_\_\_$ .

16. (This question is worth 12 points) Let the sequence $\left\{ a _ { n } \right\} ( n = 1,2,3 \ldots )$ have the sum of the first $n$ terms $S _ { n } = 2 a _ { n } - a _ { 1 }$, and $a _ { 1 } , a _ { 1 } + 1 , a _ { 3 }$ form an arithmetic sequence. (1) Find the general term formula of the sequence $\left\{ a _ { n } \right\}$; (2) Let $T _ { n }$ be the sum of the first $n$ terms of the sequence $\left\{ \frac { 1 } { a _ { n } } \right\}$. Find $T _ { n }$.

16. Let the sequence $\left\{ a _ { n } \right\}$ have the sum of the first $n$ terms $S _ { n } = 2 a _ { n } - a _ { 3 }$, and $a _ { 1 }$, $a _ { 2 } + 1$, $a _ { 3 }$ form an arithmetic sequence.
(1) Find the general term formula of the sequence $\left\{ a _ { n } \right\}$;
(2) Let $T _ { n }$ be the sum of the first $n$ terms of the sequence $\left\{ \frac { 1 } { a _ { n } } \right\}$. Find the minimum value of $n$ such that $\left| T _ { n } - 1 \right| < \frac { 1 } { 1000 }$.

17. (15 points) Given sequences $\left\{ a _ { n } \right\}$ and $\left\{ b _ { n } \right\}$ satisfying $a _ { 1 } = 2 , b _ { 1 } = 1 , a _ { n + 1 } = 2 a _ { n } \left( \mathrm { n } \in \mathrm { N } ^ { * } \right)$ , $b _ { 1 } + \frac { 1 } { 2 } b _ { 2 } + \frac { 1 } { 3 } b _ { 3 } + \cdots + \frac { 1 } { n } b _ { n } = b _ { n + 1 } - 1 \left( \mathrm { n } \in \mathrm { N } ^ { * } \right)$ .
(1) Find $a _ { n }$ and $b _ { n }$ ;
(2) Let $T _ { n }$ denote the sum of the first n terms of the sequence $\left\{ a _ { n } b _ { n } \right\}$ . Find $T _ { n }$ .

Given the sequence $\{a_n\}$ satisfies $a_{n+2} = qa_n$ (where q is a real number and $q \neq 1$), $n \in \mathbb{N}^*$, $a_1 = 1$, $a_2 = 2$, and $a_2 + a_3$, $a_3 + a_4$, $a_4 + a_5$ form an arithmetic sequence.
(I) Find the value of q and the general term formula of $\{a_n\}$;
(II) Let $b_n = \frac{\log_2 a_{2n}}{a_{2n-1}}$, $n \in \mathbb{N}^*$. Find the sum of the first n terms of the sequence $\{b_n\}$.

19. (This question is worth 13 points) Let $S _ { n }$ be the sum of the first $n$ terms of the sequence $\left\{ a _ { n } \right\}$. Given $a _ { 1 } = 1 , a _ { 2 } = 2$, and $a _ { n + 2 } = 3 S _ { n } - S _ { n + 1 } , n \in \mathbb { N } ^ { * }$. (I) Prove that: $a _ { n + 2 } = 3 a _ { n }$ (II) Find $\mathrm { S } _ { \mathrm { n } }$