Question 159
A progressão aritmética $(a_n)$ tem primeiro termo $a_1 = 3$ e razão $r = 4$. O valor de $a_{10}$ é
(A) 35 (B) 39 (C) 40 (D) 43 (E) 47

A sequência $(a_n)$ é uma progressão aritmética com $a_1 = 3$ e razão $r = 4$. O valor de $a_{10}$ é
(A) 35 (B) 39 (C) 40 (D) 43 (E) 47

The Sun's magnetic activity cycle has a period of 11 years. The beginning of the first recorded cycle occurred at the beginning of 1755 and extended until the end of 1765. Since then, all cycles of the Sun's magnetic activity have been recorded.
In the year 2101, the Sun will be in the magnetic activity cycle number
(A) 32. (B) 34. (C) 33. (D) 35. (E) 31.

In an arithmetic progression, the first term is 3 and the common difference is 4. What is the 10th term of this progression?
(A) 35
(B) 39
(C) 43
(D) 47
(E) 51

For an arithmetic sequence $\left\{ a _ { n } \right\}$ $$a _ { 1 } + a _ { 2 } = 10 , \quad a _ { 3 } + a _ { 4 } + a _ { 5 } = 45$$ When this holds, what is the value of $a _ { 10 }$? [2 points]
(1) 47
(2) 45
(3) 43
(4) 41
(5) 39

For an arithmetic sequence $\left\{ a _ { n } \right\}$, $$a _ { 5 } = 4 a _ { 3 } , \quad a _ { 2 } + a _ { 4 } = 4$$ When these conditions hold, what is the value of $a _ { 6 }$? [2 points]
(1) 5
(2) 8
(3) 11
(4) 13
(5) 16

For an arithmetic sequence $\left\{ a _ { n } \right\}$ with $a _ { 2 } = 3 , a _ { 5 } = 24$, find the value of $a _ { 7 }$. [3 points]

For an arithmetic sequence $\left\{ a _ { n } \right\}$ with common difference 2,
$$a _ { 1 } + a _ { 5 } + a _ { 9 } = 45$$
Find the value of $a _ { 1 } + a _ { 10 }$. [3 points]

For an arithmetic sequence $\left\{ a_n \right\}$, $$a_2 = 16, \quad a_5 = 10$$ Find the value of $k$ that satisfies $a_k = 0$. [3 points]

For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 2, when $a _ { 9 } = 3 a _ { 3 }$, what is the value of $a _ { 5 }$? [3 points]
(1) 10
(2) 11
(3) 12
(4) 13
(5) 14

For a sequence $\left\{ a _ { n } \right\}$, if the sum of the first $n$ terms $S _ { n } = \frac { n } { n + 1 }$, what is the value of $a _ { 4 }$? [3 points]
(1) $\frac { 1 } { 22 }$
(2) $\frac { 1 } { 20 }$
(3) $\frac { 1 } { 18 }$
(4) $\frac { 1 } { 16 }$
(5) $\frac { 1 } { 14 }$

For an arithmetic sequence $\left\{ a _ { n } \right\}$ satisfying $\sum _ { k = 1 } ^ { n } a _ { 2 k - 1 } = 3 n ^ { 2 } + n$, what is the value of $a _ { 8 }$? [4 points]
(1) 16
(2) 19
(3) 22
(4) 25
(5) 28

An arithmetic sequence $\left\{ a _ { n } \right\}$ with positive common difference satisfies the following conditions. What is the value of $a _ { 2 }$? [4 points] (가) $a _ { 6 } + a _ { 8 } = 0$ (나) $\left| a _ { 6 } \right| = \left| a _ { 7 } \right| + 3$
(1) - 15
(2) - 13
(3) - 11
(4) - 9
(5) - 7

An arithmetic sequence $\left\{ a _ { n } \right\}$ satisfies $$a _ { 5 } + a _ { 13 } = 3 a _ { 9 } , \quad \sum _ { k = 1 } ^ { 18 } a _ { k } = \frac { 9 } { 2 }$$ Find the value of $a _ { 13 }$. [4 points]
(1) 2
(2) 1
(3) 0
(4) $-1$
(5) $-2$

For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 4, $$a _ { 10 } - a _ { 7 } = 6$$ What is the value of $a _ { 4 }$? [3 points]
(1) 10
(2) 11
(3) 12
(4) 13
(5) 14

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

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. $6$

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

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 $S _ { n }$ 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

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.

If $100$ times the $100^{\text{th}}$ term of an AP with non-zero common difference equals the $50$ times its $50^{\text{th}}$ term, then the $150^{\text{th}}$ term of this AP is
(1) $-150$
(2) 150 times its $50^{\text{th}}$ term
(3) 150
(4) zero

Let $\alpha$ and $\beta$ be the roots of equation $p x ^ { 2 } + q x + r = 0 , p \neq 0$. If $p , q , r$ are in A.P. and $\frac { 1 } { \alpha } + \frac { 1 } { \beta } = 4$, then the value of $| \alpha - \beta |$ is
(1) $\frac { \sqrt { 34 } } { 9 }$
(2) $\frac { 2 \sqrt { 13 } } { 9 }$
(3) $\frac { \sqrt { 61 } } { 9 }$
(4) $\frac { 2 \sqrt { 17 } } { 9 }$

Given an A.P. whose terms are all positive integers. The sum of its first nine terms is greater than 200 and less than 220. If the second term in it is 12, then its $4^{\text{th}}$ term is:
(1) 8
(2) 24
(3) 20
(4) 16