19. Let the common difference of an arithmetic sequence be d, the sum of the first n terms be, the common ratio of a geometric sequence be q. Given $= - = 2 , \mathrm { q } = \mathrm { d } , $ $= 100$. (I) Find the general term formulas of the sequences and. (II) When $\mathrm { d } > 1$, let $= c _ { n } = \frac { a _ { n } } { b _ { n } }$. Find the sum of the first n terms of the sequence.

20. Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 }$ be terms of an arithmetic sequence with positive terms and common difference $\mathrm { d } ( d \neq 0 )$
(1) Prove that $2 ^ { a _ { 1 } } , 2 ^ { a _ { 2 } } , 2 ^ { a _ { 3 } } , 2 ^ { a _ { 4 } }$ form a geometric sequence in order
(2) Do there exist $a _ { 1 } , d$ such that $a _ { 1 } , a _ { 2 } { } ^ { 2 } , a _ { 3 } { } ^ { 3 } , a _ { 4 } { } ^ { 4 }$ form a geometric sequence in order? Explain your reasoning
(3) Do there exist $a _ { 1 } , d$ and positive integers $n , k$ such that $a _ { 1 } { } ^ { n } , a _ { 2 } { } ^ { n + k } , a _ { 3 } { } ^ { n + 3 k } , a _ { 4 } { } ^ { n + 5 k }$ form a geometric sequence in order? Explain your reasoning

Supplementary Problems

For an arithmetic sequence $\left\{ a _ { n } \right\}$, the sum of the first 9 terms is 27, and $a _ { 10 } = 8$, then $a _ { 100 } =$
(A) 100
(B) 99
(C) 98
(D) 97

Let $S$ denote the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$. If $a _ { 4 } + a _ { 5 } = 24$ and $S _ { 6 } = 48$, then the common difference of $\left\{ a _ { n } \right\}$ is
A. 1
B. 2
C. 4
D. 8

15. For an arithmetic sequence $\left\{ a _ { n } \right\}$ with sum of first $n$ terms $S _ { n }$, if $a _ { 3 } = \frac { 3 } { 2 } , S _ { 4 } = 10$, then $\sum_{k=1}^{n} \frac { 1 } { S _ { k } } = $ ______ [Figure]
III. Solving Problems: Questions 16-23 [Figure] [Figure]

(12 points)
Let $\{a_n\}$ be a sequence with $a_1 + a_2 = 2$.
(1) If $\{a_n\}$ is an arithmetic sequence and $a_1 + a_3 = 5$, find the general formula for $\{a_n\}$.
(2) If $\{a_n\}$ is a geometric sequence and $T_n$ denotes the sum of the first $n$ terms of another related sequence with $T_n = 21$, find $S_n$.

Let $S _ { n }$ denote the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$. Given $3 S _ { 3 } = S _ { 2 } + S _ { 4 }$ and $a _ { 1 } = 2$, then $a _ { 5 } =$
A. $- 12$
B. $- 10$
C. 10
D. 12

Let $S _ { n }$ be the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$. Given $a _ { 1 } = - 7 , S _ { 3 } = - 15$.
(1) Find the general term formula for $\left\{ a _ { n } \right\}$;
(2) Find $S _ { n }$ and the minimum value of $S _ { n }$.

(12 points)
Let $S _ { n }$ be the sum of the first $n$ terms of arithmetic sequence $\{ a _ { n } \}$. Given $a _ { 1 } = - 7 , S _ { 3 } = - 15$.
(1) Find the general term formula of $\{ a _ { n } \}$;
(2) Find $S _ { n }$ and the minimum value of $S _ { n }$.

8. The figure shows a flowchart for computing $\frac { 1 } { 2 + \frac { 1 } { 2 + \frac { 1 } { 2 } } }$. The blank box should be filled with
A. $A = \frac { 1 } { 2 + A }$
B. $A = 2 + \frac { 1 } { A }$
C. $A = \frac { 1 } { 1 + 2 A }$
D. $A = 1 + \frac { 1 } { 2 A }$ [Figure]

9. Let $S _ { n }$ denote the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$ . Given $S _ { 4 } = 0 , a _ { 5 } = 5$ , then
A. $a _ { n } = 2 n - 5$
B. $a _ { n } = 3 n - 10$
C. $S _ { n } = 2 n ^ { 2 } - 8 n$
D. $S _ { n } = \frac { 1 } { 2 } n ^ { 2 } - 2 n$

Let $\left\{ a _ { n } \right\}$ be an arithmetic sequence with the sum of the first $n$ terms being $S _ { n }$. If $a _ { 2 } = - 3 , S _ { 5 } = - 10$, then $a _ { 5 } =$ $\_\_\_\_$, and the minimum value of $S _ { n }$ is $\_\_\_\_$.

Let $S _ { n }$ denote the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$. If $a _ { 1 } \neq 0 , a _ { 2 } = 3 a _ { 1 }$ , then $\frac { S _ { 10 } } { S _ { 5 } } =$ \_\_\_\_\_\_.

14. Let $S _ { n }$ denote the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$ . If $a _ { 3 } = 5 , a _ { 7 } = 13$ , then $S _ { 10 } =$ $\_\_\_\_$ .

14. Let $S _ { n }$ denote the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$ . If $a _ { 1 } \neq 0$ and $a _ { 2 } = 3 a _ { 1 }$ , then $\frac { S _ { 10 } } { S _ { 5 } } =$ $\_\_\_\_$ .

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 $\left\{ \frac { 1 } { a _ { n } } + 2 n \right\}$.

18. (12 points)
Given that $\left\{ a _ { n } \right\}$ is a geometric sequence with all positive terms, $a _ { 1 } = 2 , a _ { 3 } = 2 a _ { 2 } + 16$.
(1) Find the general term formula for $\left\{ a _ { n } \right\}$;
(2) Let $b _ { n } = \log _ { 2 } a _ { n }$, find the sum of the first $n$ terms of the sequence $\left\{ b _ { n } \right\}$.

19. (12 points)
Given sequences $\left\{ a _ { n } \right\}$ and $\left\{ b _ { n } \right\}$ satisfying $a _ { 1 } = 1 , b _ { 1 } = 0,4 a _ { n + 1 } = 3 a _ { n } - b _ { n } + 4,4 b _ { n + 1 } = 3 b _ { n } - a _ { n } - 4$.
(1) Prove that $\left\{ a _ { n } + b _ { n } \right\}$ is a geometric sequence and $\left\{ a _ { n } - b _ { n } \right\}$ is an arithmetic sequence Therefore, the polar equation of the locus of point $P$ is $\rho = 4 \cos \theta , \theta \in \left[ \frac { \pi } { 4 } , \frac { \pi } { 2 } \right]$ .

The Circular Mound Altar at the Beijing Temple of Heaven is an ancient place for worshipping heaven, divided into three levels: upper, middle, and lower. At the center of the upper level is a circular stone slab (called the Heaven's Heart Stone), surrounded by 9 fan-shaped stone slabs forming the first ring, with each outer ring increasing by 9 slabs. On the next level, the first ring has 9 more slabs than the last ring of the upper level, and each outer ring also increases by 9 slabs. It is known that each level has the same number of rings, and the lower level has 729 more slabs than the middle level. The total number of fan-shaped stone slabs (excluding the Heaven's Heart Stone) in all three levels is
A． 3699 slabs
B． 3474 slabs
C． 3402 slabs
D． 3339 slabs

Given that $\left\{ a _ { n } \right\}$ is an arithmetic sequence with non-zero common difference, and $a _ { 1 } + a _ { 10 } = a _ { 9 }$, find $\frac { a _ { 1 } + a _ { 2 } + \cdots a _ { 9 } } { a _ { 10 } } =$ $\_\_\_\_$

The sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { n + 2 } + ( - 1 ) ^ { n } a _ { n } = 3 n - 1$ . The sum of the first 16 terms is 540. Then $a _ { 1 } =$ $\_\_\_\_$.

14. $x = - \frac { 3 } { 2 }$
Solution: The focus has coordinates $F \left( \frac { p } { 2 } , 0 \right)$, and point $P$ has coordinates $P \left( \frac { p } { 2 } , p \right)$. Thus $| P F | = p , | O F | = \frac { p } { 2 }$. By the focal chord property $| P F | ^ { 2 } = | O F | \cdot | F Q |$, we get $p ^ { 2 } = 3 p$, so $p = 3$. The directrix equation is $x = - \frac { 3 } { 2 }$.
15. 1
Solution: When $x \geq \frac { 1 } { 2 }$, $f ( x ) = 2 x - 2 \ln ( x ) - 1$, and $f ^ { \prime } ( x ) = \frac { 2 ( x - 1 ) } { x } > 0$, so $f ( x ) \geq f ( 1 ) = 1$. When $x < \frac { 1 } { 2 }$, $f ( x ) = 1 - 2 \ln ( x ) - 2 x$, and $f ^ { \prime } ( x ) = \frac { - 2 ( x + 1 ) } { x } < 0$, so $f ( x ) > f \left( \frac { 1 } { 2 } \right) = 2 \ln 2 > 1$. Therefore, the minimum value of function $f ( x )$ is 1.

17. Let $S _ { n }$ denote the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$ with non-zero common difference. If $a _ { 3 } = S _ { 5 }$ and $a _ { 2 } a _ { 4 } = S _ { 4 }$ .
(1) Find the general term formula $a _ { n }$ of the sequence $\left\{ a _ { n } \right\}$ ;
(2) Find the minimum value of $n$ such that $S _ { n } > a _ { n }$ holds.
Answer: (1) $a _ { n } = 2 n - 6$ ; (2) 7 .

[Solution]

[Analysis] (1) From the given conditions, first find the value of $a _ { 3 }$ , then combine with the given conditions to find the common difference of the sequence to determine the general term formula;
(2) First find the expression for the sum of the first $n$ terms, then solve the quadratic inequality to determine the minimum value of $n$ . [Detailed Solution] (1) By the properties of arithmetic sequences, we have $S _ { 5 } = 5 a _ { 3 }$ , thus: $a _ { 3 } = 5 a _ { 3 }$ , therefore $a _ { 3 } = 0$ ,
Let the common difference of the arithmetic sequence be $d$ . Then: $a _ { 2 } a _ { 4 } = \left( a _ { 3 } - d \right) \left( a _ { 3 } + d \right) = - d ^ { 2 }$ , $S _ { 4 } = a _ { 1 } + a _ { 2 } + a _ { 3 } + a _ { 4 } = \left( a _ { 3 } - 2 d \right) + \left( a _ { 3 } - d \right) + a _ { 3 } + \left( a _ { 3 } + d \right) = 4a_3 - 2d = -2d$ ,
Thus: $- d ^ { 2 } = - 2 d$ . Since the common difference is non-zero, we have: $d = 2$ ,
The general term formula of the sequence is: $a _ { n } = a _ { 3 } + ( n - 3 ) d = 2 n - 6$ .
(2) From the general term formula, we have: $a _ { 1 } = 2 - 6 = - 4$ , thus: $S _ { n } = n \times ( - 4 ) + \frac { n ( n - 1 ) } { 2 } \times 2 = n ^ { 2 } - 5 n$ . The inequality $S _ { n } > a _ { n }$ becomes: $n ^ { 2 } - 5 n > 2 n - 6$ . Simplifying: $( n - 1 ) ( n - 6 ) > 0$ ,
Solving: $n < 1$ or $n > 6$ . Since $n$ is a positive integer, the minimum value of $n$ is 7 . [Key Point] The solution of basic quantities in arithmetic sequences is a fundamental problem in arithmetic sequences. The key to solving such problems is to master the relevant formulas of arithmetic sequences and apply them flexibly.

18. Let $S _ { n }$ denote the sum of the first $n$ terms of $\left\{ a _ { n } \right\}$. Given that $a _ { n } > 0 , a _ { 2 } = 3 a _ { 1 }$, and the sequence $\left\{ \sqrt { S _ { n } } \right\}$ is an arithmetic sequence. Prove that $\left\{ a _ { n } \right\}$ is an arithmetic sequence.

18. (12 points) Given that all terms of the sequence $\{a_n\}$ are positive numbers, and $S_n$ denotes the sum of the first $n$ terms of $\{a_n\}$. Choose two of the following three statements as conditions and prove the remaining one.
(1) The sequence $\{a_n\}$ is an arithmetic sequence;
(2) The sequence $\{\sqrt{S_n}\}$ is an arithmetic sequence;
(3) $a_2 = 3a_1$.
Note: If different combinations are answered correctly, only the first answer will be scored.