For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 3, if $\sum _ { k = 1 } ^ { 5 } a _ { k } = 55$, find the value of $\sum _ { k = 1 } ^ { 5 } k \left( a _ { k } - 3 \right)$. [3 points]

For an arithmetic sequence $\left\{ a _ { n } \right\}$, $$a _ { 2 } = 6 , \quad a _ { 4 } + a _ { 6 } = 36$$ what is the value of $a _ { 10 }$? [3 points]
(1) 30
(2) 32
(3) 34
(4) 36
(5) 38

For a sequence $\left\{ a _ { n } \right\}$, $$\sum _ { k = 1 } ^ { 10 } a _ { k } - \sum _ { k = 1 } ^ { 7 } \frac { a _ { k } } { 2 } = 56 , \quad \sum _ { k = 1 } ^ { 10 } 2 a _ { k } - \sum _ { k = 1 } ^ { 8 } a _ { k } = 100$$ find the value of $a _ { 8 }$. [3 points]

An arithmetic sequence $\left\{ a _ { n } \right\}$ with all positive terms and equal first term and common difference satisfies $$\sum _ { k = 1 } ^ { 15 } \frac { 1 } { \sqrt { a _ { k } } + \sqrt { a _ { k + 1 } } } = 2$$ What is the value of $a _ { 4 }$? [3 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

For two sequences $\left\{ a _ { n } \right\}$ and $\left\{ b _ { n } \right\}$, $$\sum _ { k = 1 } ^ { 5 } \left( 3 a _ { k } + 5 \right) = 55 , \quad \sum _ { k = 1 } ^ { 5 } \left( a _ { k } + b _ { k } \right) = 32$$ What is the value of $\sum _ { k = 1 } ^ { 5 } b _ { k }$? [3 points]

For an arithmetic sequence $\{a_n\}$ with nonzero common difference, $$|a_6| = a_8, \quad \sum_{k=1}^{5} \frac{1}{a_k a_{k+1}} = \frac{5}{96}$$ Find the value of $\sum_{k=1}^{15} a_k$. [4 points]
(1) 60
(2) 65
(3) 70
(4) 75
(5) 80

For two sequences $\{a_n\}$ and $\{b_n\}$, $$\sum_{k=1}^{10} a_k = \sum_{k=1}^{10} (2b_k - 1), \quad \sum_{k=1}^{10} (3a_k + b_k) = 33$$ Find the value of $\sum_{k=1}^{10} b_k$. [3 points]

A sequence $\left\{ a_{n} \right\}$ with $a_{1} = 2$ and an arithmetic sequence $\left\{ b_{n} \right\}$ with $b_{1} = 2$ satisfy $$\sum_{k=1}^{n} \frac{a_{k}}{b_{k+1}} = \frac{1}{2}n^{2}$$ for all natural numbers $n$. What is the value of $\sum_{k=1}^{5} a_{k}$? [4 points]
(1) 120
(2) 125
(3) 130
(4) 135
(5) 140

A sequence $\left\{ a_{n} \right\}$ satisfies $$a_{n} + a_{n+4} = 12$$ for all natural numbers $n$. What is the value of $\sum_{n=1}^{16} a_{n}$? [3 points]

All terms of a sequence $\left\{ a_{n} \right\}$ are integers and satisfy the following conditions. What is the sum of the values of $\left| a_{1} \right|$? [4 points] (가) For all natural numbers $n$, $$a_{n+1} = \begin{cases} a_{n} - 3 & \left(\left| a_{n} \right| \text{ is odd}\right) \\ \frac{1}{2}a_{n} & \left(a_{n} = 0 \text{ or } \left| a_{n} \right| \text{ is even}\right) \end{cases}$$ (나) The minimum value of the natural number $m$ such that $\left| a_{m} \right| = \left| a_{m+2} \right|$ is 3.

For the sequence $\left\{ a _ { n } \right\}$, when $\sum _ { k = 1 } ^ { 4 } \left( 2 a _ { k } - k \right) = 0$, what is the value of $\sum _ { k = 1 } ^ { 4 } a _ { k }$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

23. (Total: 18 points; Part 1: 4 points; Part 2: 6 points; Part 3: 8 points) Given that the general terms of sequences $\{a_n\}$ and $\{b_n\}$ are $a_n = 3n + 6, b_n = 2n + 7$ $(n \in \mathbb{N}^*)$ respectively. The elements in the set $\{x \mid x = a_n, n \in \mathbb{N}^*\} \cup \{x \mid x = b_n, n \in \mathbb{N}^*\}$ are arranged in increasing order to form a sequence $c_1, c_2, c_3, \cdots, c_n, \cdots$
(1) Find the three smallest numbers that are both terms of sequence $\{a_n\}$ and terms of sequence $\{b_n\}$;
(2) Among the terms $c_1, c_2, c_3, \cdots, c_{40}$, how many are not terms of sequence $\{b_n\}$? Please explain your reasoning;
(3) Find the sum of the first $4n$ terms $S_{4n}$ $(n \in \mathbb{N}^*)$ of sequence $\{c_n\}$.
2011 Shanghai College Entrance Examination Mathematics (Liberal Arts) Answers
1. $\{x \mid x < 1\}$;
2. $-2$;
3. $-\frac{3}{2}$;
4. $\sqrt{5}$;
5. $x + 2y - 11 = 0$; 6. $x < 0$ or $x > 1$; 7. $3\pi$; 8. $\sqrt{6}$; 9. $\frac{5}{2}$; 10. 2; 11. 6; 12. $\frac{15}{2}$; 13. 0.985; 14. $[-2, 7]$.
15. A; 16. D; 17. A; 18. B. 19. Solution: $(z_1 - 2)(1 + i) = 1 - i \Rightarrow z_1 = 2 - i$
Let $z_2 = a + 2i, a \in \mathbb{R}$, then $z_1 z_2 = (2 - i)(a + 2i) = (2a + 2) + (4 - a)i$. Since $z_1 z_2 \in \mathbb{R}$, we have $4 - a = 0$, so $z_2 = 4 + 2i$.
20. Solution: (1) Connect $BD, AB_1, B_1D_1, AD_1$. Since $BD \parallel B_1D_1$ and $AB_1 = AD_1$, the angle between skew lines $BD$ and $AB_1$ is $\angle AB_1D_1$. Let $\angle AB_1D_1 = \theta$. $\cos \theta = \frac{AB_1^2 + B_1D_1^2 - AD_1^2}{2AB_1 \times B_1D_1} = \frac{\sqrt{10}}{10}$. Therefore, the angle between skew lines $BD$ and $AB_1$ is $\arccos \frac{\sqrt{10}}{10}$.
(2) Connect $AC, CB_1, CD_1$. The volume of the tetrahedron is [Figure] $V = V_{ABCD - A_1B_1C_1D_1} - 4 \times V_{C - B_1C_1D_1} = 2 - 4 \times \frac{1}{3} = \frac{2}{3}$. 21. Solution: (1) When $a > 0, b > 0$, for any $x_1, x_2 \in \mathbb{R}, x_1 < x_2$, we have $f(x_1) - f(x_2) = a(2^{x_1} - 2^{x_2}) + b(3^{x_1} - 3^{x_2})$. Since $2^{x_1} < 2^{x_2}, a > 0 \Rightarrow a(2^{x_1} - 2^{x_2}) < 0$, and $3^{x_1} < 3^{x_2}, b > 0 \Rightarrow b(3^{x_1} - 3^{x_2}) < 0$, we have $f(x_1) - f(x_2) < 0$, so function $f(x)$ is increasing on $\mathbb{R}$.
When $a < 0, b < 0$, by similar reasoning, function $f(x)$ is decreasing on $\mathbb{R}$.
(2) $f(x+1) - f(x) = a \cdot 2^x + 2b \cdot 3^x > 0$. When $a < 0, b > 0$, we have $\left(\frac{3}{2}\right)^x > -\frac{a}{2b}$, so $x > \log_{1.5}\left(-\frac{a}{2b}\right)$. When $a > 0, b < 0$, we have $\left(\frac{3}{2}\right)^x < -\frac{a}{2b}$, so $x < \log_{1.5}\left(-\frac{a}{2b}\right)$.
22. Solution: (1) $m = 2$, the ellipse equation is $\frac{x^2}{4} + y^2 = 1$, and $

2. In the arithmetic sequence $\left\{ a _ { n } \right\}$, if $a _ { 2 } = 4 , a _ { 4 } = 2$, then $a _ { 6 } =$
A. $-1$
B. $0$
C. $1$
D. $6$

3. Execute the flowchart shown in Figure 1. If the input is $n = 3$, then the output is
A. $\frac { 6 } { 7 }$
B. $\frac { 3 } { 7 }$
C. $\frac { 8 } { 9 }$
D. $\frac { 4 } { 9 }$ [Figure]
(4) If variables $x , y$ satisfy the constraint conditions $\left\{ \begin{array} { c } x + y \geq - 1 \\ 2 x - y \leq 1 \\ y \leq 1 \end{array} \right.$, then the minimum value of $z = 3 x - y$ is
(A) $- 7$
(B) $- 1$
(C) $1$
(D) $2$
(5) Let the function $f ( x ) = \ln ( 1 + x ) - \ln ( 1 - x )$. Then $f ( x )$ is
(A) an odd function and increasing on $( 0,1 )$
(B) an odd function and decreasing on $(0,1)$
(C) an even function and increasing on $( 0,1 )$
(D) an even function and decreasing on $(0,1)$ (6) Given that the expansion of $\left( \sqrt { \mathrm { x } } - \frac { \mathrm { a } } { \sqrt { \mathrm { x } } } \right) ^ { 5 }$ contains a term with $\mathrm { x } ^ { \frac { 3 } { 2 } }$ whose coefficient is 30, then $\mathrm { a } =$
(A) $\sqrt { 3 }$
(B) $- \sqrt { 3 }$
(C) $6$
(D) $- 6$ (7) In the square shown in Figure 2, 10000 points are randomly thrown. The estimated number of points falling in the shaded region (where curve C is the density curve of the normal distribution $N ( 0,1 )$) is
[Figure]
Figure 2
(A) $2386$
(B) $2718$
(C) $3413$
(D) $4772$
Attachment: If $X \sim N \left( \mu , \sigma ^ { 2 } \right)$, then
$$\begin{aligned} & P ( \mu - \sigma < x \leq \mu + \sigma ) = 0.6826 \\ & P ( \mu - 2 \sigma < x \leq \mu + 2 \sigma ) = 0.9544 \end{aligned}$$
(8) Points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ move on the circle $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } = 1$, and $\mathrm { AB } \perp \mathrm { BC }$. If point P has coordinates $( 2,0 )$, then the maximum value of $| \overrightarrow { P A } + \overrightarrow { P B } + \overrightarrow { P C } |$ is
A. $6$
B. $7$
C. $8$
D. $9$ (9) The graph of the function $f ( x ) = \sin 2 x$ is shifted to the right by $\varphi \left( 0 < \varphi < \frac { \pi } { 2 } \right)$ units to obtain the graph of function $g ( x )$. If for $x _ { 1 }$ and $x _ { 2 }$ satisfying $\left| f \left( x _ { 1 } \right) - g \left( x _ { 2 } \right) \right| = 2$, we have $\left| x _ { 1 } - x _ { 2 } \right| _ { \min } = \frac { \pi } { 3 }$, then $\varphi =$
A. $\frac { 5 \pi } { 12 }$
B. $\frac { \pi } { 3 }$
C. $\frac { \pi } { 4 }$
D. $\frac { \pi } { 6 }$

3. Given that $\left\{ a _ { n } \right\}$ is an arithmetic sequence with non-zero common difference $d$, and the sum of the first $n$ terms is $S _ { n }$. If $a _ { 3 } , a _ { 4 } , a _ { 8 }$ form a geometric sequence, then
A. $a _ { 1 } d > 0 , d S _ { n } > 0$
B. $a _ { 1 } d < 0 , d S _ { n } < 0$
C. $a _ { 1 } d > 0 , d S _ { n } < 0$
D. $a _ { 1 } d < 0 , d S _ { n } > 0$ [Figure]

5. Executing the flowchart shown in Figure 2, if the input is $n = 3$, then the output $S =$
A. $\frac { 6 } { 7 }$
B. $\frac { 3 } { 7 }$
C. $\frac { 8 } { 9 }$
D. $\frac { 4 } { 9 }$
[Figure]
Figure 2

5. Let $S _ { n }$ be the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$. If $a _ { 1 } + a _ { 3 } + a _ { 5 } = 3$, then $S _ { 5 } =$ [Figure] [Figure]
A. $5$
B. $7$
C. $9$
D. $11$

6. Let $\left\{ a _ { n } \right\}$ be an arithmetic sequence. The correct conclusion is
A. If $a _ { 1 } + a _ { 2 } > 0$, then $a _ { 2 } + a _ { 3 } > 0$
B. If $a _ { 1 } + a _ { 3 } < 0$, then $a _ { 1 } + a _ { 2 } < 0$
C. If $0 < a _ { 1 } < a _ { 2 }$, then $a _ { 2 } > \sqrt { a _ { 1 } a _ { 3 } }$
D. If $a _ { 1 } < 0$, then $\left( a _ { 2 } - a _ { 1 } \right) \left( a _ { 2 } - a _ { 3 } \right) > 0$

10. Given that $\left\{ a _ { n } \right\}$ is an arithmetic sequence with non-zero common difference $d$. If $a _ { 2 } , a _ { 3 } , a _ { 7 }$ form a geometric sequence, and $2 a _ { 1 } + a _ { 2 } = 1$ , then $a _ { 1 } =$ $\_\_\_\_$ , $d =$ $\_\_\_\_$.

11. The sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } = 1$ and $a _ { n + 1 } - a _ { n } = n + 1 \quad \left( n \in N ^ { * } \right)$, then the sum of the first 10 terms of the sequence $\left\{ \frac { 1 } { a _ { n } } \right\}$ is $\_\_\_\_$.

13. A group of 1010 numbers with median 1010 form an arithmetic sequence with last term 2015. The first term of this sequence is $\_\_\_\_$

Given that the arithmetic sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } + a _ { 2 } = 10$ and $a _ { 4 } - a _ { 3 } = 2$.\n(I) Find the general term formula of $\left\{ a _ { n } \right\}$;\n(II) Let the geometric sequence $\left\{ b _ { n } \right\}$ satisfy $b _ { 2 } = a _ { 3 }$ and $b _ { 3 } = a _ { 7 }$. Question: Which term of the sequence $\left\{ a _ { n } \right\}$ is equal to $b _ { 6 }$?

An arithmetic sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 3 } = 2$ and the sum of the first 3 terms $S _ { 3 } = \frac { 9 } { 2 }$ .
(I) Find the general term formula for $\left\{ a _ { n } \right\}$;
(II) A geometric sequence $\left\{ b _ { n } \right\}$ satisfies $b _ { 1 } = a _ { 1 } , b _ { 4 } = a _ { 15 }$. Find the sum of the first $n$ terms $T _ { n }$ of $\left\{ b _ { n } \right\}$.

With the development of China's economy, residents' savings deposits have increased year by year. The table below shows the year-end balance of urban and rural residents' RMB savings deposits in a certain region:

Year	2010	2011	2012	2013	2014
Time code $t$	1	2	3	4	5
\begin{tabular}{ l } Savings deposits $y$
(hundred billion yuan)

& 5 & 6 & 7 & 8 & 10 \hline \end{tabular}
(I) Find the regression equation $\hat { y } = \hat { b } t + \hat { a }$ for $y$ with respect to $t$.
(II) Use the regression equation to predict the RMB savings deposits in this region for 2015 ($t = 6$). Note: In the regression equation $\hat { y } = \hat { b } t + \hat { a }$,
$$\hat { b } = \frac { \sum _ { i = 1 } ^ { n } t _ { i } y _ { i } - n \overline { t } \overline { y } } { \sum _ { i = 1 } ^ { n } t _ { i } ^ { 2 } - n \bar { t } ^ { 2 } } , \hat { \mathrm { a } } = \overline { \mathrm { y } } - \hat { \mathrm { b } } \overline { \mathrm { t } }$$

18. Given that $\{ a _ { n } \}$ is a geometric sequence with all positive terms, $\{ b _ { n } \}$ is an arithmetic sequence, and $a _ { 1 } = b _ { 1 } = 1$, $b _ { 2 } + b _ { 3 } = 2 a _ { 3 }$, $a _ { 5 } - 3 b _ { 2 } = 7$.
(1) Find the general term formulas for $\{ a _ { n } \}$ and $\{ b _ { n } \}$;
(2) Let $c _ { n } = a _ { n } b _ { n } , n \in \mathbb{N} ^ { * }$. Find the sum of the first $n$ terms of the sequence $\{ c _ { n } \}$.