The sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } = 1$, and with $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$, $$a _ { n + 1 } = ( n + 1 ) S _ { n } + n ! \quad ( n \geq 1 )$$ The following is the process of finding the general term $a _ { n }$.
For a natural number $n$, since $a _ { n + 1 } = S _ { n + 1 } - S _ { n }$, by the given equation, $$S _ { n + 1 } = ( n + 2 ) S _ { n } + n ! \quad ( n \geq 1 )$$ Dividing both sides by $( n + 2 ) !$, $$\frac { S _ { n + 1 } } { ( n + 2 ) ! } = \frac { S _ { n } } { ( n + 1 ) ! } + \frac { 1 } { ( n + 1 ) ( n + 2 ) }$$ Let $b _ { n } = \frac { S _ { n } } { ( n + 1 ) ! }$. Then $b _ { 1 } = \frac { 1 } { 2 }$ and $$b _ { n + 1 } = b _ { n } + \frac { 1 } { ( n + 1 ) ( n + 2 ) }$$ Finding the general term of the sequence $\left\{ b _ { n } \right\}$, $$b _ { n } = \frac { ( \text{(가)} ) } { n + 1 }$$ Therefore, $$S _ { n } = \text{(가)} \times n!$$ Thus, $$a _ { n } = \text{(나)} \times ( n - 1 ) ! \quad ( n \geq 1 )$$ When the expressions that fit (가) and (나) are $f ( n )$ and $g ( n )$ respectively, what is the value of $f ( 7 ) + g ( 6 )$? [4 points]
(1) 44
(2) 41
(3) 38
(4) 35
(5) 32

For a natural number $k$, $$a _ { k } = \lim _ { n \rightarrow \infty } \frac { \left( \frac { 6 } { k } \right) ^ { n + 1 } } { \left( \frac { 6 } { k } \right) ^ { n } + 1 }$$ Find the value of $\sum _ { k = 1 } ^ { 10 } k a _ { k }$. [4 points]

As shown in the figure, for a square ABCD with side length 5, let the five division points of diagonal BD be $\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } , \mathrm { P } _ { 4 }$ in order from point B. Draw squares with diagonals $\mathrm { BP } _ { 1 } , \mathrm { P } _ { 2 } \mathrm { P } _ { 3 } , \mathrm { P } _ { 4 } \mathrm { D }$ and circles with diameters $\mathrm { P } _ { 1 } \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } \mathrm { P } _ { 4 }$, then color the figure-eight-shaped region to obtain figure $R _ { 1 }$. In figure $R _ { 1 }$, let $\mathrm { Q } _ { 1 }$ be the vertex of the square with diagonal $\mathrm { P } _ { 2 } \mathrm { P } _ { 3 }$ closest to point A, and $\mathrm { Q } _ { 2 }$ be the vertex closest to point C. Draw squares with diagonals $\mathrm { AQ } _ { 1 }$ and $\mathrm { CQ } _ { 2 }$, and in these 2 new squares, draw figure-eight-shaped figures using the same method as for $R _ { 1 }$ and color them to obtain figure $R _ { 2 }$. In figure $R _ { 2 }$, in the squares with diagonals $\mathrm { AQ } _ { 1 }$ and $\mathrm { CQ } _ { 2 }$, draw figure-eight-shaped figures using the same method as obtaining $R _ { 2 }$ from $R _ { 1 }$ and color them to obtain figure $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the area of the colored region in the $n$-th figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [3 points]
(1) $\frac { 24 } { 17 } ( \pi + 3 )$
(2) $\frac { 25 } { 17 } ( \pi + 3 )$
(3) $\frac { 26 } { 17 } ( \pi + 3 )$
(4) $\frac { 24 } { 17 } ( 2 \pi + 1 )$
(5) $\frac { 25 } { 17 } ( 2 \pi + 1 )$

A sequence $\left\{ a _ { n } \right\}$ with all positive terms has $a _ { 1 } = a _ { 2 } = 1$, and with $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$, it satisfies $$a _ { n + 1 } = \frac { S _ { n } ^ { 2 } } { S _ { n - 1 } } + ( 2 n - 1 ) S _ { n } \quad ( n \geq 2 )$$ The following is the process of finding the general term $a _ { n }$. $$\begin{gathered} \text{Since } a _ { n + 1 } = S _ { n + 1 } - S _ { n } \text{, from the given equation we have} \\ S _ { n + 1 } = \frac { S _ { n } ^ { 2 } } { S _ { n - 1 } } + 2 n S _ { n } \quad ( n \geq 2 ) \end{gathered}$$ Dividing both sides by $S _ { n }$, we get $$\frac { S _ { n + 1 } } { S _ { n } } = \frac { S _ { n } } { S _ { n - 1 } } + 2 n$$ Let $b _ { n } = \frac { S _ { n + 1 } } { S _ { n } }$. Then $b _ { 1 } = 2$ and $$b _ { n } = b _ { n - 1 } + 2 n \quad ( n \geq 2 )$$ The general term of sequence $\left\{ b _ { n } \right\}$ is $$b _ { n } = \text { (가) } \times ( n + 1 ) \quad ( n \geq 1 )$$ Therefore, $$S _ { n } = ( \text{가} ) \times \{ ( n - 1 ) ! \} ^ { 2 } \quad ( n \geq 1 )$$ Thus $a _ { 1 } = 1$, and for $n \geq 2$, $$\begin{aligned} a _ { n } & = S _ { n } - S _ { n - 1 } \\ & = \text { (나) } \times \{ ( n - 2 ) ! \} ^ { 2 } \end{aligned}$$ Let $f ( n )$ and $g ( n )$ be the expressions that fit (가) and (나), respectively. What is the value of $f ( 10 ) + g ( 6 )$? [4 points]
(1) 110
(2) 125
(3) 140
(4) 155
(5) 170

A sequence $\left\{ a _ { n } \right\}$ with all positive terms has $a _ { 1 } = a _ { 2 } = 1$, and when $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$, $$a _ { n + 1 } = \frac { S _ { n } ^ { 2 } } { S _ { n - 1 } } + ( 2 n - 1 ) S _ { n } \quad ( n \geq 2 )$$ The following is the process of finding the general term $a _ { n }$.
Since $a _ { n + 1 } = S _ { n + 1 } - S _ { n }$, from the given equation we have $$S _ { n + 1 } = \frac { S _ { n } ^ { 2 } } { S _ { n - 1 } } + 2 n S _ { n } \quad ( n \geq 2 )$$ Dividing both sides by $S _ { n }$, $$\frac { S _ { n + 1 } } { S _ { n } } = \frac { S _ { n } } { S _ { n - 1 } } + 2 n$$ Let $b _ { n } = \frac { S _ { n + 1 } } { S _ { n } }$. Then $b _ { 1 } = 2$ and $$b _ { n } = b _ { n - 1 } + 2 n \quad ( n \geq 2 )$$ Finding the general term of the sequence $\left\{ b _ { n } \right\}$, $$b _ { n } = \text { (가) } \times ( n + 1 ) \quad ( n \geq 1 )$$ Therefore, $$S _ { n } = ( \text{가} ) \times \{ ( n - 1 ) ! \} ^ { 2 } \quad ( n \geq 1 )$$ Thus $a _ { 1 } = 1$, and for $n \geq 2$, $$\begin{aligned} a _ { n } & = S _ { n } - S _ { n - 1 } \\ & = \text { (나) } \times \{ ( n - 2 ) ! \} ^ { 2 } \end{aligned}$$ When the expressions for (가) and (나) are $f ( n )$ and $g ( n )$ respectively, what is the value of $f ( 10 ) + g ( 6 )$? [4 points]
(1) 110
(2) 125
(3) 140
(4) 155
(5) 170

As shown in the figure, there is a circle $O$ with diameter AB of length 4. Let C be the center of the circle, and let D and P be the midpoints of segments AC and BC, respectively. Let E and Q be the points where the perpendicular bisector of segment AC and the perpendicular bisector of segment BC meet the upper semicircle of circle $O$, respectively. Draw a square DEFG with side DE that meets circle $O$ at point A and has diagonal DF, and draw a square PQRS with side PQ that meets circle $O$ at point B and has diagonal PR. Color the $\square$-shaped figure that is the common part of the interior of circle $O$ and the interior of square DEFG, and the $\square$-shaped figure that is the common part of the interior of circle $O$ and the interior of square PQRS to obtain figure $R _ { 1 }$. In figure $R _ { 1 }$, draw circle $O _ { 1 }$ centered at point F with radius $\frac { 1 } { 2 } \overline { \mathrm { DE } }$, and circle $O _ { 2 }$ centered at point R with radius $\frac { 1 } { 2 } \overline { \mathrm { PQ } }$. Color 2 $\square$-shaped figures and 2 $\square$-shaped figures created in the same way as obtaining figure $R _ { 1 }$ on the two circles $O _ { 1 }$ and $O _ { 2 }$ to obtain figure $R _ { 2 }$. Continuing this process, let $S _ { n }$ be the area of the colored part in figure $R _ { n }$ obtained the $n$-th time. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 12 \pi - 9 \sqrt { 3 } } { 10 }$
(2) $\frac { 8 \pi - 6 \sqrt { 3 } } { 5 }$
(3) $\frac { 32 \pi - 24 \sqrt { 3 } } { 15 }$
(4) $\frac { 28 \pi - 21 \sqrt { 3 } } { 10 }$
(5) $\frac { 16 \pi - 12 \sqrt { 3 } } { 5 }$

For a natural number $n$, let $\mathrm { P } _ { n }$ be the point where the line $x = 4 ^ { n }$ meets the curve $y = \sqrt { x }$. Let $L _ { n }$ be the length of the segment $\mathrm { P } _ { n } \mathrm { P } _ { n + 1 }$. Find the value of $\lim _ { n \rightarrow \infty } \left( \frac { L _ { n + 1 } } { L _ { n } } \right) ^ { 2 }$. [4 points]

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

A function $f ( x )$ satisfies $\lim _ { x \rightarrow 1 } ( x + 1 ) f ( x ) = 1$. When $\lim _ { x \rightarrow 1 } \left( 2 x ^ { 2 } + 1 \right) f ( x ) = a$, find the value of $20 a$. [3 points]

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

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

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

What is the value of $\lim _ { n \rightarrow \infty } \frac { \frac { 5 } { n } + \frac { 3 } { n ^ { 2 } } } { \frac { 1 } { n } - \frac { 2 } { n ^ { 3 } } }$?

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]

13. Let $s _ { n }$ be the sum of the first n terms of the geometric sequence $\left\{ a _ { n } \right\}$. If $a _ { 1 } = 1$ and $3 s _ { 1 } , 2 s _ { 2 } , s _ { 3 }$ form an arithmetic sequence, then $a _ { n } = $ $\_\_\_\_$.

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 $\left| T _ { n } - 1 \right| < \frac { 1 } { 1000 }$.