A soma dos termos de uma progressão aritmética finita com 10 termos, primeiro termo igual a 2 e último termo igual a 20, é
(A) 100 (B) 110 (C) 120 (D) 130 (E) 140

The projections for rice production in the period 2012-2021, in a certain producing region, point to a perspective of constant growth in annual production. The table presents the quantity of rice, in tons, that will be produced in the first years of this period, according to this projection.

Year	Production projection (t)
2012	50.25
2013	51.50
2014	52.75
2015	54.00

The total amount of rice, in tonnes, that should be produced in the period from 2012 to 2021 will be
(A) 497.25. (B) 500.85. (C) 502.87. (D) 558.75. (E) 563.25.

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.

QUESTION 160
The sum of the first 10 terms of an arithmetic progression with first term 2 and common difference 3 is
(A) 145
(B) 155
(C) 165
(D) 175
(E) 185

In a school project, João was invited to calculate the areas of several different squares, arranged in sequence, from left to right, as shown in the figure.
The first square in the sequence has a side measuring 1 cm, the second square has a side measuring 2 cm, the third square has a side measuring 3 cm, and so on. The objective of the project is to identify by how much the area of each square in the sequence exceeds the area of the previous square. The area of the square that occupies position $n$ in the sequence was represented by $\mathrm{A}_{n}$.
For $n \geq 2$, the value of the difference $\mathrm{A}_{n} - \mathrm{A}_{n-1}$, in square centimeter, is equal to
(A) $2n - 1$
(B) $2n + 1$
(C) $-2n + 1$
(D) $(n-1)^{2}$
(E) $n^{2} - 1$

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

The value of $\displaystyle\sum_{k=1}^{5} k^2$ is:
(A) 45
(B) 50
(C) 55
(D) 60
(E) 65

The sum $$S = 1 + 111 + 11111 + \cdots + \underbrace{11\cdots1}_{2k+1}$$ is equal to . . . . . . .

Suppose $a_0, a_1, a_2, a_3, \ldots$ is an arithmetic progression with $a_0$ and $a_1$ positive integers. Let $g_0, g_1, g_2, g_3, \ldots$ be the geometric progression such that $g_0 = a_0$ and $g_1 = a_1$.
Statements
(1) We must have $\left(a_5\right)^2 \geq a_0 a_{10}$.
(2) The sum $a_0 + a_1 + \cdots + a_{10}$ must be a multiple of the integer $a_5$.
(3) If $\sum_{i=0}^{\infty} a_i$ is $+\infty$ then $\sum_{i=0}^{\infty} g_i$ is also $+\infty$.
(4) If $\sum_{i=0}^{\infty} g_i$ is finite then $\sum_{i=0}^{\infty} a_i$ is $-\infty$.

The three sides of triangle $a < b < c$ are in arithmetic progression (AP) with common difference $d = b - a = c - b$. Denote the angles opposite to sides $a , b , c$ respectively by $A , B , C$.
Statements
(1) $d$ must be less than $a$.
(2) $d$ can be any positive number less than $a$.
(3) The numbers $\sin A , \sin B , \sin C$ are in AP.
(4) The numbers $\cos A , \cos B , \cos C$ are in AP.

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 two points $\mathrm { P } ( n , f ( n ) )$ and $\mathrm { Q } ( n + 1 , f ( n + 1 ) )$ on the graph of the quadratic function $f ( x ) = 3 x ^ { 2 }$, let $a _ { n }$ be the distance between them. Find the value of $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { n }$. (Here, $n$ is a natural number.) [4 points]
(1) 9
(2) 8
(3) 7
(4) 6
(5) 5

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

Two sequences $\left\{ a _ { n } \right\} , \left\{ b _ { n } \right\}$ are given by
$$\begin{aligned} & a _ { n } = \frac { 1 } { 2 ^ { n - 1 } } \cos \frac { ( n - 1 ) \pi } { 2 } \\ & b _ { n } = \frac { 1 + ( - 1 ) ^ { n - 1 } } { 2 ^ { n } } \end{aligned}$$
Which of the following statements in are true? [4 points]

ㄱ. For all natural numbers $k$, $a _ { 3 k } < 0$. ㄴ. For all natural numbers $k$, $a _ { 4 k - 1 } + b _ { 4 k - 1 } = 0$. ㄷ. $\sum _ { n = 1 } ^ { \infty } a _ { n } = \frac { 3 } { 5 } \sum _ { n = 1 } ^ { \infty } b _ { n }$
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄴ
(5) ㄴ, ㄷ

In the sequence $\left\{ a _ { n } \right\}$, $a _ { 1 } = 1 , a _ { 2 } = 4 , a _ { 3 } = 10$, and the sequence $\left\{ a _ { n + 1 } - a _ { n } \right\}$ is a geometric sequence. Find the value of $a _ { 5 }$. [3 points]

Three numbers $a , 0 , b$ form an arithmetic sequence in this order, and three numbers $2 b , a , - 7$ form a geometric sequence in this order. What is the value of $a$? [3 points]
(1) 10
(2) 12
(3) 14
(4) 16
(5) 18

For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 0 and nonzero common difference, the sequence $\left\{ b _ { n } \right\}$ satisfies $a _ { n + 1 } b _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$. Find the value of $b _ { 27 }$. [4 points]

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]

An arithmetic sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 2 } + a _ { 4 } = 8$ and $a _ { 7 } = 52$. Find the common difference. [3 points]

For a natural number $n$, point $\mathrm { A } _ { n }$ is a point on the $x$-axis. Point $\mathrm { A } _ { n + 1 }$ is determined according to the following rule. (가) The coordinates of point $\mathrm { A } _ { 1 }$ are $( 2,0 )$. (나) (1) Let $\mathrm { P } _ { n }$ be the point where the line passing through point $\mathrm { A } _ { n }$ and parallel to the $y$-axis meets the curve $y = \frac { 1 } { x } ( x > 0 )$.
(2) Let $\mathrm { Q } _ { n }$ be the point obtained by reflecting point $\mathrm { P } _ { n }$ about the line $y = x$.
(3) Let $\mathrm { R } _ { n }$ be the point where the line passing through point $\mathrm { Q } _ { n }$ and parallel to the $y$-axis meets the $x$-axis.
(4) Let $\mathrm { A } _ { n + 1 }$ be the point obtained by translating point $\mathrm { R } _ { n }$ by 1 unit in the direction of the $x$-axis. Let the $x$-coordinate of point $\mathrm { A } _ { n }$ be $x _ { n }$. When $x _ { 5 } = \frac { q } { p }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [3 points]

For a natural number $n$, point $\mathrm { A } _ { n }$ is on the $x$-axis. Point $\mathrm { A } _ { n + 1 }$ is determined according to the following rule. (가) The coordinates of point $\mathrm { A } _ { 1 }$ are $( 2,0 )$. (나) (1) Let $\mathrm { P } _ { n }$ be the point where the line passing through point $\mathrm { A } _ { n }$ and parallel to the $y$-axis meets the curve $y = \frac { 1 } { x } ( x > 0 )$.
(2) Let $\mathrm { Q } _ { n }$ be the point obtained by reflecting $\mathrm { P } _ { n }$ about the line $y = x$.
(3) Let $\mathrm { R } _ { n }$ be the point where the line passing through $\mathrm { Q } _ { n }$ and parallel to the $y$-axis meets the $x$-axis.

A sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { n + 1 } - a _ { n } = 2 n$. When $a _ { 10 } = 94$, what is the value of $a _ { 1 }$? [3 points]
(1) 5
(2) 4
(3) 3
(4) 2
(5) 1

For a sequence $\left\{ a _ { n } \right\}$, let $S _ { n }$ denote the sum of the first $n$ terms. The sequence $\left\{ S _ { 2 n - 1 } \right\}$ is an arithmetic sequence with common difference $-3$, and the sequence $\left\{ S _ { 2 n } \right\}$ is an arithmetic sequence with common difference $2$. When $a _ { 2 } = 1$, find the value of $a _ { 8 }$. [4 points]

There is a rectangle $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ with $\overline { \mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } } = 1$ and $\overline { \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 } } = 2$. As shown in the figure, let $\mathrm { M } _ { 1 }$ be the midpoint of segment $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$, and on segment $\mathrm { A } _ { 1 } \mathrm { D } _ { 1 }$, two points $\mathrm { B } _ { 2 } , \mathrm { C } _ { 2 }$ are determined such that $\angle \mathrm { A } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 2 } = \angle \mathrm { C } _ { 2 } \mathrm { M } _ { 1 } \mathrm { D } _ { 1 } = 15 ^ { \circ } , \angle \mathrm { B } _ { 2 } \mathrm { M } _ { 1 } \mathrm { C } _ { 2 } = 60 ^ { \circ }$. Let $S _ { 1 }$ be the sum of the area of triangle $\mathrm { A } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 2 }$ and the area of triangle $\mathrm { C } _ { 2 } \mathrm { M } _ { 1 } \mathrm { D } _ { 1 }$.
Quadrilateral $\mathrm { A } _ { 2 } \mathrm {~B} _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ is a rectangle with $\overline { \mathrm { B } _ { 2 } \mathrm { C } _ { 2 } } = 2 \overline { \mathrm {~A} _ { 2 } \mathrm {~B} _ { 2 } }$ such that two points $\mathrm { A } _ { 2 } , \mathrm { D } _ { 2 }$ are determined as shown in the figure. Let $\mathrm { M } _ { 2 }$ be the midpoint of segment $\mathrm { B } _ { 2 } \mathrm { C } _ { 2 }$, and on segment $\mathrm { A } _ { 2 } \mathrm { D } _ { 2 }$, two points $\mathrm { B } _ { 3 } , \mathrm { C } _ { 3 }$ are determined such that $\angle \mathrm { A } _ { 2 } \mathrm { M } _ { 2 } \mathrm {~B} _ { 3 } = \angle \mathrm { C } _ { 3 } \mathrm { M } _ { 2 } \mathrm { D } _ { 2 } = 15 ^ { \circ }$, $\angle \mathrm { B } _ { 3 } \mathrm { M } _ { 2 } \mathrm { C } _ { 3 } = 60 ^ { \circ }$. Let $S _ { 2 }$ be the sum of the area of triangle $\mathrm { A } _ { 2 } \mathrm { M } _ { 2 } \mathrm {~B} _ { 3 }$ and the area of triangle $\mathrm { C } _ { 3 } \mathrm { M } _ { 2 } \mathrm { D } _ { 2 }$. Continuing this process, what is the value of $\sum _ { n = 1 } ^ { \infty } S _ { n }$ for the obtained $S _ { n }$? [4 points]
(1) $\frac { 2 + \sqrt { 3 } } { 6 }$
(2) $\frac { 3 - \sqrt { 3 } } { 2 }$
(3) $\frac { 4 + \sqrt { 3 } } { 9 }$
(4) $\frac { 5 - \sqrt { 3 } } { 5 }$
(5) $\frac { 7 - \sqrt { 3 } } { 8 }$