Let $a_{n}$, $n \geq 1$ be a sequence of real numbers. If $a_{n} \rightarrow a$, show that
$$b_{n} = \frac{a_{1} + 2a_{2} + 3a_{3} + \cdots + na_{n}}{n^{2}} \rightarrow \frac{a}{2}.$$

Let $a _ { 0 }$ and $a _ { 1 }$ be complex numbers and define $a _ { n } = 2 a _ { n - 1 } + a _ { n - 2 }$ for $n \geq 2$.
(A) Show that there are polynomials $p ( z ) , q ( z ) \in \mathbb { C } [ z ]$ such that $q ( 0 ) \neq 0$ and $\sum _ { n \geq 0 } a _ { n } z ^ { n }$ is the Taylor series expansion (around 0) of $\frac { p ( z ) } { q ( z ) }$.
(B) Let $a _ { 0 } = 1$ and $a _ { 1 } = 2$. Show that there exist complex numbers $\beta _ { 1 } , \beta _ { 2 } , \gamma _ { 1 } , \gamma _ { 2 }$ such that $$a _ { n } = \beta _ { 1 } \gamma _ { 1 } ^ { n + 1 } + \beta _ { 2 } \gamma _ { 2 } ^ { n + 1 }$$ for all $n$.

[12 points] Let $f$ be a function from natural numbers to natural numbers that satisfies
$$\begin{aligned} & f ( n ) = n - 2 \quad \text { for } n > 3000 \\ & f ( n ) = f ( f ( n + 5 ) ) \quad \text { for } n \leq 3000 \end{aligned}$$
Show that $f ( 2022 )$ is uniquely decided and find its value.

What is the value of $\lim _ { n \rightarrow \infty } \left( \sqrt { n ^ { 2 } + 6 n + 4 } - n \right)$? [3 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) 1
(4) 2
(5) 3

For a sequence $\left\{ a _ { n } \right\}$ where the sum of the first $n$ terms $S _ { n } = 2 n + \frac { 1 } { 2 ^ { n } }$, what is the value of $\lim _ { n \rightarrow \infty } a _ { n }$? [3 points]
(1) 2
(2) 1
(3) $\frac { 1 } { 2 }$
(4) $\frac { 1 } { 4 }$
(5) 0

When a sequence $\left\{ a _ { n } \right\}$ satisfies $n < a _ { n } < n + 1$ for all natural numbers $n$, what is the value of $\lim _ { n \rightarrow \infty } \frac { n ^ { 2 } } { a _ { 1 } + a _ { 2 } + \cdots + a _ { n } }$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Find the value of $\lim _ { n \rightarrow \infty } \frac { 5 \cdot 3 ^ { n + 1 } - 2 ^ { n + 1 } } { 3 ^ { n } + 2 ^ { n } }$. [3 points]

What is the value of $\lim _ { n \rightarrow \infty } \frac { 3 + \left( \frac { 1 } { 3 } \right) ^ { n } } { 2 + \left( \frac { 1 } { 2 } \right) ^ { n } }$? [2 points]
(1) 1
(2) $\frac { 3 } { 2 }$
(3) 2
(4) $\frac { 5 } { 2 }$
(5) 3

For a sequence $\left\{ a _ { n } \right\}$ with $a _ { 1 } = 2$ and $a _ { n + 1 } = 2 a _ { n } + 2$, what is the value of $a _ { 10 }$? [3 points]
(1) 1022
(2) 1024
(3) 2021
(4) 2046
(5) 2082

For a sequence $\left\{ a _ { n } \right\}$ with $\sum _ { n = 1 } ^ { \infty } \frac { a _ { n } } { 4 ^ { n } } = 2$, find the value of $\lim _ { n \rightarrow \infty } \frac { a _ { n } + 4 ^ { n + 1 } - 3 ^ { n - 1 } } { 4 ^ { n - 1 } + 3 ^ { n + 1 } }$. [3 points]

What is the value of $\lim _ { n \rightarrow \infty } \frac { 2 } { \sqrt { n ^ { 2 } + 2 n } - \sqrt { n ^ { 2 } + 1 } }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Let $a_n$ denote the sum of all natural numbers such that when divided by a natural number $n$ ($n \geqq 2$), the quotient and remainder are equal. For example, when divided by 4, the natural numbers for which the quotient and remainder are equal are $5, 10, 15$, so $a_4 = 5 + 10 + 15 = 30$. Find the minimum value of the natural number $n$ satisfying $a_n > 500$. [4 points]

What is the value of $\lim _ { n \rightarrow \infty } \frac { ( n + 1 ) ( 3 n - 1 ) } { 2 n ^ { 2 } + 1 }$? [2 points]
(1) $\frac { 3 } { 2 }$
(2) 2
(3) $\frac { 5 } { 2 }$
(4) 3
(5) $\frac { 7 } { 2 }$

[Discrete Mathematics] A sequence $\left\{ a _ { n } \right\}$ satisfies $$\left\{ \begin{array} { l } a _ { 1 } = 2 , a _ { 2 } = 5 \\ a _ { n } = 2 a _ { n - 1 } + a _ { n - 2 } \end{array} \quad ( n \geqq 3 ) \right.$$ What is the value of $a _ { 5 }$? [3 points]
(1) 70
(2) 72
(3) 74
(4) 76
(5) 78

The sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } = 1$ and $$a _ { n + 1 } = n + 1 + \frac { ( n - 1 ) ! } { a _ { 1 } a _ { 2 } \cdots a _ { n } } \quad ( n \geqq 1 )$$ The following is part of the process of finding the general term $a _ { n }$.
For all natural numbers $n$, $$a _ { 1 } a _ { 2 } \cdots a _ { n } a _ { n + 1 } = a _ { 1 } a _ { 2 } \cdots a _ { n } \times ( n + 1 ) + ( n - 1 ) !$$ If $b _ { n } = \frac { a _ { 1 } a _ { 2 } \cdots a _ { n } } { n ! }$, then $b _ { 1 } = 1$ and $$b _ { n + 1 } = b _ { n } + ( \text{(a)} )$$ The general term of the sequence $\left\{ b _ { n } \right\}$ is $b _ { n } =$ (b) so $\frac { a _ { 1 } a _ { 2 } \cdots a _ { n } } { n ! } =$ (b). $\vdots$ Therefore, $a _ { 1 } = 1$ and $a _ { n } = \frac { ( n - 1 ) ( 2 n - 1 ) } { 2 n - 3 }$ for $n \geqq 2$.
When the expression that fits (a) is $f ( n )$ and the expression that fits (b) is $g ( n )$, what is the value of $f ( 13 ) \times g ( 7 )$? [4 points]
(1) $\frac { 1 } { 70 }$
(2) $\frac { 1 } { 77 }$
(3) $\frac { 1 } { 84 }$
(4) $\frac { 1 } { 91 }$
(5) $\frac { 1 } { 98 }$

The sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } = 1$ and $$a _ { n + 1 } = n + 1 + \frac { ( n - 1 ) ! } { a _ { 1 } a _ { 2 } \cdots a _ { n } } \quad ( n \geqq 1 )$$ The following is part of the process of finding the general term $a _ { n }$.
For all natural numbers $n$, $$a _ { 1 } a _ { 2 } \cdots a _ { n } a _ { n + 1 } = a _ { 1 } a _ { 2 } \cdots a _ { n } \times ( n + 1 ) + ( n - 1 ) !$$ If $b _ { n } = \frac { a _ { 1 } a _ { 2 } \cdots a _ { n } } { n ! }$, then $b _ { 1 } = 1$ and $$b _ { n + 1 } = b _ { n } + ( \text{(가)} )$$ The general term of the sequence $\left\{ b _ { n } \right\}$ is $b _ { n } =$ (나) so $\frac { a _ { 1 } a _ { 2 } \cdots a _ { n } } { n ! } =$ (나). $\vdots$ Therefore, $a _ { 1 } = 1$ and $a _ { n } = \frac { ( n - 1 ) ( 2 n - 1 ) } { 2 n - 3 } ( n \geqq 2 )$.
When the expression that fits (가) is $f ( n )$ and the expression that fits (나) is $g ( n )$, what is the value of $f ( 13 ) \times g ( 7 )$? [4 points]
(1) $\frac { 1 } { 70 }$
(2) $\frac { 1 } { 77 }$
(3) $\frac { 1 } { 84 }$
(4) $\frac { 1 } { 91 }$
(5) $\frac { 1 } { 98 }$

What is the value of $\lim _ { n \rightarrow \infty } \frac { 5 ^ { n + 1 } + 2 } { 5 ^ { n } + 3 ^ { n } }$? [2 points]
(1) 2
(2) 3
(3) 4
(4) 5
(5) 6

For a sequence $\left\{ a _ { n } \right\}$ with first term 1, let $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$. The following holds:
$$n S _ { n + 1 } = ( n + 2 ) S _ { n } + ( n + 1 ) ^ { 3 } \quad ( n \geq 1 )$$
The following is part of the process of finding the general term of the sequence $\left\{ a _ { n } \right\}$.
Since $S _ { n + 1 } = S _ { n } + a _ { n + 1 }$ for natural numbers $n$,
$$n a _ { n + 1 } = 2 S _ { n } + ( n + 1 ) ^ { 3 } \quad \cdots (\text{ㄱ})$$
For natural numbers $n \geq 2$,
$$( n - 1 ) a _ { n } = 2 S _ { n - 1 } + n ^ { 3 } \quad \cdots (\text{ㄴ})$$
Subtracting (ㄴ) from (ㄱ), we obtain
$$n a _ { n + 1 } = ( n + 1 ) a _ { n } + \text{ (A) }$$
Dividing both sides by $n ( n + 1 )$,
$$\frac { a _ { n + 1 } } { n + 1 } = \frac { a _ { n } } { n } + \frac { \text{ (A) } } { n ( n + 1 ) }$$
Let $b _ { n } = \frac { a _ { n } } { n }$. Then
$$b _ { n + 1 } = b _ { n } + 3 + \text{ (B) } \quad ( n \geq 2 )$$
Therefore
$$b _ { n } = b _ { 2 } + \text{ (C) } \quad ( n \geq 3 )$$
holds.
What are the correct expressions for (A), (B), and (C)?

For a sequence $\left\{ a _ { n } \right\}$ with first term 1, let $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$. The following relation holds:
$$n S _ { n + 1 } = ( n + 2 ) S _ { n } + ( n + 1 ) ^ { 3 } \quad ( n \geq 1 )$$
The following is part of the process of finding the general term of the sequence $\left\{ a _ { n } \right\}$.
Since $S _ { n + 1 } = S _ { n } + a _ { n + 1 }$ for natural numbers $n$,
$$n a _ { n + 1 } = 2 S _ { n } + ( n + 1 ) ^ { 3 } \quad \cdots (\text{ㄱ})$$
For natural numbers $n \geq 2$,
$$( n - 1 ) a _ { n } = 2 S _ { n - 1 } + n ^ { 3 } \quad \cdots (\text{ㄴ})$$
By subtracting (ㄴ) from (ㄱ), we obtain
$$n a _ { n + 1 } = ( n + 1 ) a _ { n } + \quad \text { (가) }$$
Dividing both sides by $n ( n + 1 )$,
$$\frac { a _ { n + 1 } } { n + 1 } = \frac { a _ { n } } { n } + \frac { ( \text{가} ) } { n ( n + 1 ) }$$
If $b _ { n } = \frac { a _ { n } } { n }$, then
$$b _ { n + 1 } = b _ { n } + \text{ (나) } \quad ( n \geq 2 )$$
so
$$b _ { n } = b _ { 2 } + \text{ (다) } \quad ( n \geq 3 )$$
When the expressions that go in (가), (나), and (다) are $f ( n ) , g ( n ) , h ( n )$ respectively, what is the value of $\frac { f ( 3 ) } { g ( 3 ) h ( 6 ) }$? [4 points]
(1) 30
(2) 36
(3) 42
(4) 48
(5) 54

What is the value of $\lim _ { n \rightarrow \infty } \frac { 5n ^ { 2 } + 1 } { 3n ^ { 2 } - 1 }$? [2 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 2 } { 3 }$
(3) 1
(4) $\frac { 4 } { 3 }$
(5) $\frac { 5 } { 3 }$

For the sequence $\left\{ a_n \right\}$, $$\sum_{n=1}^{\infty} \left( na_n - \frac{n^2 + 1}{2n + 1} \right) = 3$$ What is the value of $\lim_{n \rightarrow \infty} \left( a_n^2 + 2a_n + 2 \right)$? [4 points]
(1) $\frac{13}{4}$
(2) 3
(3) $\frac{11}{4}$
(4) $\frac{5}{2}$
(5) $\frac{9}{4}$

What is the value of $\lim _ { n \rightarrow \infty } \frac { 2 \times 3 ^ { n + 1 } + 5 } { 3 ^ { n } }$? [2 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

A sequence $\left\{ a _ { n } \right\}$ with all positive terms has $a _ { 1 } = 10$ and satisfies $$\left( a _ { n + 1 } \right) ^ { n } = 10 \left( a _ { n } \right) ^ { n + 1 } \quad ( n \geq 1 )$$ The following is the process of finding the general term $a _ { n }$.
Taking the common logarithm of both sides of the given equation, $$n \log a _ { n + 1 } = ( n + 1 ) \log a _ { n } + 1$$ Dividing both sides by $n ( n + 1 )$, $$\frac { \log a _ { n + 1 } } { n + 1 } = \frac { \log a _ { n } } { n } + ( \text { (가) } )$$ Let $b _ { n } = \frac { \log a _ { n } } { n }$. Then $b _ { 1 } = 1$ and $$b _ { n + 1 } = b _ { n } + \text { (가) }$$ Finding the general term of the sequence $\left\{ b _ { n } \right\}$, $$b _ { n } = \text { (나) }$$ Therefore, $$\log a _ { n } = n \times \text { (나) }$$ Thus $a _ { n } = 10 ^ { n \times ( \text { (나) } ) }$.
When the expressions for (가) and (나) are $f ( n )$ and $g ( n )$ respectively, what is the value of $\frac { g ( 10 ) } { f ( 4 ) }$? [3 points]
(1) 38
(2) 40
(3) 42
(4) 44
(5) 46

In rectangle $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$, $\overline { \mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } } = 1 , \overline { \mathrm {~A} _ { 1 } \mathrm { D } _ { 1 } } = 2$. As shown in the figure, let $\mathrm { M } _ { 1 } , \mathrm {~N} _ { 1 }$ be the midpoints of segments $\mathrm { A } _ { 1 } \mathrm { D } _ { 1 }$ and $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$ respectively.
Draw a circular sector $\mathrm { N } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$ with center $\mathrm { N } _ { 1 }$, radius $\overline { \mathrm { B } _ { 1 } \mathrm {~N} _ { 1 } }$, and central angle $\frac { \pi } { 2 }$, and draw a circular sector $\mathrm { D } _ { 1 } \mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$ with center $\mathrm { D } _ { 1 }$, radius $\overline { \mathrm { C } _ { 1 } \mathrm { D } _ { 1 } }$, and central angle $\frac { \pi } { 2 }$.
The region enclosed by the arc $\mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$ and segment $\mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$ of sector $\mathrm { N } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$ and the region enclosed by the arc $\mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$ and segment $\mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$ of sector $\mathrm { D } _ { 1 } \mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$ are shaded to form a checkmark shape, and the resulting figure is called $R _ { 1 }$.
In figure $R _ { 1 }$, a rectangle $\mathrm { A } _ { 2 } \mathrm {~B} _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ is drawn with vertices at point $\mathrm { A } _ { 2 }$ on segment $\mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$, point $\mathrm { D } _ { 2 }$ on arc $\mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$, and two points $\mathrm { B } _ { 2 } , \mathrm { C } _ { 2 }$ on side $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$ such that $\overline { \mathrm { A } _ { 2 } \mathrm {~B} _ { 2 } } : \overline { \mathrm { A } _ { 2 } \mathrm { D } _ { 2 } } = 1 : 2$. A checkmark shape is shaded in rectangle $\mathrm { A } _ { 2 } \mathrm {~B} _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ using the same method as for figure $R _ { 1 }$, and the resulting figure is called $R _ { 2 }$.
Continuing this process, let $S _ { n }$ be the area of the shaded region in the $n$-th figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 25 } { 19 } \left( \frac { \pi } { 2 } - 1 \right)$
(2) $\frac { 5 } { 4 } \left( \frac { \pi } { 2 } - 1 \right)$
(3) $\frac { 25 } { 21 } \left( \frac { \pi } { 2 } - 1 \right)$
(4) $\frac { 25 } { 22 } \left( \frac { \pi } { 2 } - 1 \right)$
(5) $\frac { 25 } { 23 } \left( \frac { \pi } { 2 } - 1 \right)$

A sequence $\left\{ a _ { n } \right\}$ with all positive terms has $a _ { 1 } = 10$ and satisfies
$$\left( a _ { n + 1 } \right) ^ { n } = 10 \left( a _ { n } \right) ^ { n + 1 } \quad ( n \geq 1 )$$
The following is the process of finding the general term $a _ { n }$.
Taking the common logarithm of both sides of the given equation:
$$n \log a _ { n + 1 } = ( n + 1 ) \log a _ { n } + 1$$
Dividing both sides by $n ( n + 1 )$:
$$\frac { \log a _ { n + 1 } } { n + 1 } = \frac { \log a _ { n } } { n } + ( \text{(가)} )$$
Let $b _ { n } = \frac { \log a _ { n } } { n }$. Then $b _ { 1 } = 1$ and
$$b _ { n + 1 } = b _ { n } + \text{(가)}$$
Finding the general term of the sequence $\left\{ b _ { n } \right\}$:
$$b _ { n } = \text{(나)}$$
Therefore,
$$\log a _ { n } = n \times \text{(나)}$$
Thus $a _ { n } = 10 ^ { n \times \text{(나)} }$.
Let $f ( n )$ and $g ( n )$ be the expressions that fit in (가) and (나), respectively. What is the value of $\frac { g ( 10 ) } { f ( 4 ) }$? [4 points]
(1) 38
(2) 40
(3) 42
(4) 44
(5) 46