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]

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) $\frac { 5 } { 12 }$

For an arithmetic sequence $\left\{ a _ { n } \right\}$ with nonzero common difference, the three terms $a _ { 2 } , a _ { 4 } , a _ { 9 }$ form a geometric sequence with common ratio $r$ in this order. Find the value of $6r$. [4 points]

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 a sequence $\left\{ a _ { n } \right\}$ with $a _ { 1 } = 1$ and satisfying
$$a _ { n + 1 } = \frac { 2 n } { n + 1 } a _ { n }$$
for all natural numbers $n$, what is the value of $a _ { 4 }$? [3 points]
(1) $\frac { 3 } { 2 }$
(2) 2
(3) $\frac { 5 } { 2 }$
(4) 3
(5) $\frac { 7 } { 2 }$

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

Three numbers $a , a + b , 2 a - b$ form an arithmetic sequence in this order, and three numbers $1 , a - 1, 3 b + 1$ form a geometric sequence with positive common ratio in this order. Find the value of $a ^ { 2 } + b ^ { 2 }$. [3 points]

Three numbers $a , a + b , 2 a - b$ form an arithmetic sequence in this order, and three numbers $1 , a - 1, 3 b + 1$ form a geometric sequence with positive common ratio in this order. Find the value of $a ^ { 2 } + b ^ { 2 }$. [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 natural numbers $n$, the point $\mathrm{P}_n$ on the coordinate plane is determined according to the following rules. (가) The coordinates of the three points $\mathrm{P}_1, \mathrm{P}_2, \mathrm{P}_3$ are $(-1, 0)$, $(1, 0)$, and $(-1, 2)$, respectively. (나) The midpoint of segment $\mathrm{P}_n \mathrm{P}_{n+1}$ and the midpoint of segment $\mathrm{P}_{n+2} \mathrm{P}_{n+3}$ are the same. For example, the coordinates of point $\mathrm{P}_4$ are $(1, -2)$. If the coordinates of point $\mathrm{P}_{25}$ are $(a, b)$, find the value of $a + b$. [4 points]

For a natural number $n$, the point $\mathrm { P } _ { n }$ on the coordinate plane is determined according to the following rules.
(a) The coordinates of the three points $\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 }$ are $( - 1,0 ) , ( 1,0 )$, and $( - 1,2 )$, respectively.
(b) The midpoint of line segment $\mathrm { P } _ { n } \mathrm { P } _ { n + 1 }$ and the midpoint of line segment $\mathrm { P } _ { n + 2 } \mathrm { P } _ { n + 3 }$ are the same. For example, the coordinates of point $\mathrm { P } _ { 4 }$ are $( 1 , - 2 )$. When the coordinates of point $\mathrm { P } _ { 25 }$ are $( a , b )$, find the value of $a + b$. [4 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 an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 6 and common difference $d$, let $S _ { n }$ denote the sum of the first $n$ terms. When
$$\frac { a _ { 8 } - a _ { 6 } } { S _ { 8 } - S _ { 6 } } = 2$$
holds, what is the value of $d$? [3 points]
(1) $-1$
(2) $-2$
(3) $-3$
(4) $-4$
(5) $-5$

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) $120 \left( \log _ { 3 } 2 + 1 \right)$

A sequence $\left\{ a _ { n } \right\}$ satisfies the following conditions. (가) $a _ { 1 } = a _ { 2 } + 3$ (나) $a _ { n + 1 } = - 2 a _ { n } ( n \geq 1 )$ Find the value of $a _ { 9 }$. [3 points]

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

For an arithmetic sequence $\left\{ a _ { n } \right\}$, when $a _ { 8 } - a _ { 4 } = 28$, find the common difference of the sequence $\left\{ a _ { n } \right\}$. [3 points]

For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 2, $$2 \left( a _ { 2 } + a _ { 3 } \right) = a _ { 9 }$$ Find the common difference of the sequence $\left\{ a _ { n } \right\}$. [3 points]

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

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

Find the number of ways to partition the natural number 11 into natural numbers between 3 and 7 (inclusive). [3 points]
(1) 2
(2) 4
(3) 6
(4) 8
(5) 10

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