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.

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

A club has a soccer field with a total area of $8000 \mathrm{~m}^{2}$, corresponding to the grass. Usually, the grass mowing of this field is done by two machines owned by the club for this service. Working at the same pace, the two machines mow together $200 \mathrm{~m}^{2}$ per hour. Due to the urgency of holding a soccer match, the field administrator will need to request machines from the neighboring club equal to his own to do the mowing work in a maximum time of 5 h.
Using the two machines that the club already has, what is the minimum number of machines that the field administrator should request from the neighboring club?
(A) 4
(B) 6
(C) 8
(D) 14
(E) 16

A passion fruit producer uses a water tank with volume $V$ to feed the irrigation system of his orchard. The system draws water through a hole at the bottom of the tank at a constant flow rate. With the water tank full, the system was activated at 7 a.m. on Monday. At 1 p.m. on the same day, it was found that 15\% of the water volume in the tank had already been used. An electronic device interrupts the system's operation when the remaining volume in the tank is 5\% of the total volume, for refilling.
Assuming that the system operates without failures, at what time will the electronic device interrupt the operation?
(A) At 3 p.m. on Monday.
(B) At 11 a.m. on Tuesday.
(C) At 2 p.m. on Tuesday.
(D) At 4 a.m. on Wednesday.
(E) At 9 p.m. on Tuesday.

In a cafeteria, the success of summer sales are juices prepared based on fruit pulp. One of the best-selling juices is strawberry with acerola, which is prepared with $\frac{2}{3}$ of strawberry pulp and $\frac{1}{3}$ of acerola pulp.
For the merchant, the pulps are sold in packages of equal volume. Currently, the strawberry pulp package costs $\mathrm{R}\$ 18.00$ and the acerola one costs $\mathrm{R}\$ 14.70$. However, a price increase is expected for the acerola pulp package next month, rising to $\mathrm{R}\$ 15.30$.
To not increase the price of the juice, the merchant negotiated with the supplier a reduction in the price of the strawberry pulp package.
The reduction, in reais, in the price of the strawberry pulp package should be
(A) 1.20.
(B) 0.90.
(C) 0.60.
(D) 0.40.
(E) 0.30.

On the coordinate plane, for a natural number $n$, let $\mathrm { A } _ { n }$ be the point where the two lines $y = \frac { 1 } { n } x$ and $x = n$ meet, and let $\mathrm { B } _ { n }$ be the point where the line $x = n$ and the $x$-axis meet. Let $\mathrm { C } _ { n }$ be the center of the circle inscribed in triangle $\mathrm { A } _ { n } \mathrm { OB } _ { n }$, and let $S _ { n }$ be the area of triangle $\mathrm { A } _ { n } \mathrm { OC } _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } \frac { S _ { n } } { n }$? [4 points]
(1) $\frac { 1 } { 12 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 3 }$
(5) $\frac { 5 } { 12 }$

The sequence $\left\{ a _ { n } \right\}$ satisfies $$2 a _ { n + 1 } = a _ { n } + a _ { n + 2 }$$ for all natural numbers $n$. When $a _ { 2 } = - 1 , a _ { 3 } = 2$, what is the sum of the first 10 terms of the sequence $\left\{ a _ { n } \right\}$? [3 points]
(1) 95
(2) 90
(3) 85
(4) 80
(5) 75

For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term $- 5$ and common difference 2, what is the value of $\sum _ { k = 11 } ^ { 20 } a _ { k }$? [3 points]
(1) 260
(2) 255
(3) 250
(4) 245
(5) 240

For a natural number $n$, $f ( n )$ is defined as follows:
$$f ( n ) = \begin{cases} \log _ { 3 } n & ( n \text{ is odd} ) \\ \log _ { 2 } n & ( n \text{ is even} ) \end{cases}$$
For the sequence $\left\{ a _ { n } \right\}$ where $a _ { n } = f \left( 6 ^ { n } \right) - f \left( 3 ^ { n } \right)$, what is the value of $\sum _ { n = 1 } ^ { 15 } a _ { n }$? [3 points]
(1) $120 \left( \log _ { 2 } 3 - 1 \right)$
(2) $105 \log _ { 3 } 2$
(3) $105 \log _ { 2 } 3$
(4) $120 \log _ { 2 } 3$
(5) $120 \left( \log _ { 3 } 2 + 1 \right)$

For the function $f ( x ) = \frac { 1 } { 2 } x + 2$, find the value of $\sum _ { k = 1 } ^ { 15 } f ( 2 k )$. [3 points]

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 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 $\left\{ a_{n} \right\}$ 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]

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

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. $11$

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 $\frac { S _ { 10 } } { S _ { 5 } } =$ $\_\_\_\_$ .

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 $\frac { a _ { 1 } + a _ { 2 } + \cdots a _ { 9 } } { a _ { 10 } } =$ $\_\_\_\_$

In the sequence $\left\{ a_{n} \right\}$ , let $S_{n}$ be the sum of the first $n$ terms of $\left\{ a_{n} \right\}$ , $S_{5} = 5S_{3} - 4$ , then $S_{4} =$
A. $7$
B. $9$
C. $15$
D. $20$

Let $S _ { n }$ denote the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$. If $a _ { 3 } + a _ { 4 } = 7$ and $3 a _ { 2 } + a _ { 5 } = 5$, then $S _ { 10 } =$ $\_\_\_\_$ .

Let $S_n$ denote the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$. If $S_3 = 6$, $S_5 = -5$, then $S_6 = $ ( )
A. $-20$
B. $-15$
C. $-10$
D. $-5$

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ be an A.P, such that $\frac { a _ { 1 } + a _ { 2 } + \ldots + a _ { p } } { a _ { 1 } + a _ { 2 } + a _ { 3 } + \ldots + a _ { q } } = \frac { p ^ { 3 } } { q ^ { 3 } } ; p \neq q$. Then $\frac { a _ { 6 } } { a _ { 21 } }$ is equal to:
(1) $\frac { 41 } { 11 }$
(2) $\frac { 31 } { 121 }$
(3) $\frac { 11 } { 41 }$
(4) $\frac { 121 } { 1861 }$