The sum $1^2 - 2 \cdot 3^2 + 3 \cdot 5^2 - 4 \cdot 7^2 + 5 \cdot 9^2 - \ldots + 15 \cdot 29^2$ is $\_\_\_\_$.

Let $a _ { n }$ be $n ^ { \text {th} }$ term of the series $5 + 8 + 14 + 23 + 35 + 50 + \ldots\ldots$. and $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$. Then $S _ { 30 } - a _ { 40 }$ is equal to
(1) 11310
(2) 11260
(3) 11290
(4) 11280

The sum of the series $\frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + \cdots + \frac{1}{(2n-1)(2n+1)}$ is $\_\_\_\_$.

The sum of the series $\frac { 1 } { 1 - 3 \cdot 1 ^ { 2 } + 1 ^ { 4 } } + \frac { 2 } { 1 - 3 \cdot 2 ^ { 2 } + 2 ^ { 4 } } + \frac { 3 } { 1 - 3 \cdot 3 ^ { 2 } + 3 ^ { 4 } } +\ldots$ up to 10 terms is
(1) $\frac { 45 } { 109 }$
(2) $- \frac { 45 } { 109 }$
(3) $\frac { 55 } { 109 }$
(4) $- \frac { 55 } { 109 }$

For positive integers $n$, if $4 a _ { n } = \left( n ^ { 2 } + 5 n + 6 \right)$ and $S _ { n } = \sum _ { k = 1 } ^ { n } \left( \frac { 1 } { a _ { k } } \right)$, then the value of $507 S _ { 2025 }$ is :
(1) 540
(2) 675
(3) 1350
(4) 135

If $\sum _ { r = 1 } ^ { n } T _ { r } = \frac { ( 2 n - 1 ) ( 2 n + 1 ) ( 2 n + 3 ) ( 2 n + 5 ) } { 64 }$, then $\lim _ { n \rightarrow \infty } \sum _ { r = 1 } ^ { n } \left( \frac { 1 } { T _ { r } } \right)$ is equal to:
(1) 0
(2) $\frac { 2 } { 3 }$
(3) 1
(4) $\frac { 1 } { 3 }$

The sequence $\left\{ a _ { n } \right\}$ is defined by
$$a _ { 1 } = \frac { 2 } { 9 } , \quad a _ { n } = \frac { ( n + 1 ) ( 2 n - 3 ) } { 3 n ( 2 n + 1 ) } a _ { n - 1 } \quad ( n = 2,3,4 , \cdots ) .$$
We are to find the general term $a _ { n }$ and the infinite sum $\sum _ { n = 1 } ^ { \infty } a _ { n }$.
(1) For A $\sim$ E in the following sentences, choose the correct answers from among (0) $\sim$ (9) below.
First, when we set $b _ { n } = \frac { n + 1 } { 3 ^ { n } a _ { n } }$ and express $\frac { b _ { n } } { b _ { n - 1 } }$ in terms of $n$, we have
$$\frac { b _ { n } } { b _ { n - 1 } } = \frac { \mathbf { A } } { \mathbf { B } } \cdot \frac { a _ { n - 1 } } { a _ { n } } = \frac { \mathbf { C } } { \mathbf { D } }$$
From this equation, we have
$$a _ { n } = \frac { n + 1 } { 3 ^ { n } ( \mathbf { E } ) ( 2 n + 1 ) } .$$
(0) $n - 1$
(1) $n$
(2) $n + 1$
(3) $2 n - 1$
(4) $2 n + 1$
(5) $2 n - 3$ (6) $2 n + 3$ (7) $3 n - 1$ (8) $3 n$ (9) $3 n + 1$
(2) Next, let $c _ { n } = \frac { 1 } { 3 ^ { n } ( 2 n + 1 ) } ( n = 0,1,2 , \cdots )$. When we set $a _ { n } = A c _ { n - 1 } + B c _ { n }$, we see that $A = \frac { \mathbf { F } } { \mathbf { G } }$ and $B = \frac { \mathbf { H I } } { \mathbf { G } }$. Using this result to find $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$, we have
$$S _ { n } = \frac { \mathbf { K } } { \mathbf { L } } \left( \mathbf { M } \right)$$
Hence we obtain
$$\sum _ { n = 1 } ^ { \infty } a _ { n } = \lim _ { n \rightarrow \infty } S _ { n } = \frac { \mathbf { N } } { \mathbf { O } }$$

When an artist uses single-point perspective to draw spatial scenes on a flat piece of paper, the following principles must be followed: I. A straight line in space must be drawn as a straight line on the paper. II. The relative positions of points on a line in space must be consistent with the relative positions of the points drawn on the paper. III. The $K$ value of any four distinct points on a line in space must be the same as the $K$ value of the four points drawn on the paper, where the $K$ value is defined as follows: For any four ordered distinct points $P_1, P_2, P_3, P_4$ on a line, the corresponding $K$ value is defined as $$K = \frac{\overline{P_1P_4} \times \overline{P_2P_3}}{\overline{P_1P_3} \times \overline{P_2P_4}}$$ An artist follows the above principles to draw a line in space and four distinct points $Q_1, Q_2, Q_3, Q_4$ on that line on paper, where $\overline{Q_1Q_2} = \overline{Q_2Q_3} = \overline{Q_3Q_4}$. If the line drawn on the paper is viewed as a number line and the points on it are represented by coordinates, which of the following sets of four coordinates is most likely to be the coordinates of these four points on the paper?
(1) $1, 2, 4, 8$
(2) $3, 4, 6, 9$
(3) $1, 5, 8, 9$
(4) $1, 2, 4, 9$
(5) $1, 7, 9, 10$

Select the value of $1 . \overline { 5 } \times 5$.
(1) $7 . \overline { 5 }$
(2) $7 . \overline { 6 }$
(3) $7 . \overline { 7 }$
(4) $7 . \overline { 8 }$
(5) $7 . \overline { 9 }$

Let $f _ { 1 }$ be a positive constant function on $[ 0,1 ]$ with $f _ { 1 } ( x ) = c$, and let $p$ and $q$ be positive real numbers with $1 / p + 1 / q = 1$. Moreover, let $\left\{ f _ { n } \right\}$ be the sequence of functions on $[ 0,1 ]$ defined by
$$f _ { n + 1 } ( x ) = p \int _ { 0 } ^ { x } \left( f _ { n } ( t ) \right) ^ { 1 / q } \mathrm {~d} t \quad ( n = 1,2 , \ldots )$$
Answer the following questions.
(1) Let $\left\{ a _ { n } \right\}$ and $\left\{ c _ { n } \right\}$ be the sequences of real numbers defined by $a _ { 1 } = 0 , c _ { 1 } = c$ and
$$\begin{aligned} & a _ { n + 1 } = q ^ { - 1 } a _ { n } + 1 \quad ( n = 1,2 , \ldots ) \\ & c _ { n + 1 } = \frac { p \left( c _ { n } \right) ^ { 1 / q } } { a _ { n + 1 } } \quad ( n = 1,2 , \ldots ) \end{aligned}$$
Show that $f _ { n } ( x ) = c _ { n } x ^ { a _ { n } }$.
(2) Let $g _ { n }$ be the function on $[ 0,1 ]$ defined by $g _ { n } ( x ) = x ^ { a _ { n } } - x ^ { p }$ for $n \geq 2$. Noting that $a _ { n } \geq 1$ holds true for $n \geq 2$, show that $g _ { n }$ attains its maximum at a point $x = x _ { n }$, and find the value of $x _ { n }$.
(3) Show that $\lim _ { n \rightarrow \infty } g _ { n } ( x ) = 0$ for any $x \in [ 0,1 ]$.
(4) Let $d _ { n }$ be defined by $d _ { n } = \left( c _ { n } \right) ^ { q ^ { n } }$. Show that $d _ { n + 1 } / d _ { n }$ converges to a finite positive value as $n \rightarrow \infty$. You may use the fact that $\lim _ { t \rightarrow \infty } ( 1 - 1 / t ) ^ { t } = 1 / \mathrm { e }$.
(5) Find the value of $\lim _ { n \rightarrow \infty } c _ { n }$.
(6) Show that $\lim _ { n \rightarrow \infty } f _ { n } ( x ) = x ^ { p }$ for any $x \in [ 0,1 ]$.

$$\sum _ { n = 4 } ^ { 9 } \left( \prod _ { k = 1 } ^ { n } \frac { k + 1 } { k } \right)$$
What is the result of this operation?
A) 45
B) 48
C) 50
D) 52
E) 54

The sequence $\left( a _ { n } \right)$
$$a _ { n } = \begin{cases} 2 ^ { n } + 1 , & n \equiv 0 ( \bmod 2 ) \\ 2 ^ { n } - 1 , & n \equiv 1 ( \bmod 2 ) \end{cases}$$
is defined in the form. Accordingly, what is the value of the expression $\frac { a _ { 9 } - a _ { 7 } } { a _ { 8 } - 4 \cdot a _ { 6 } }$?
A) $-2 ^ { 8 }$
B) $-2 ^ { 7 }$
C) $-2 ^ { 6 }$
D) $-2 ^ { 5 }$
E) $-2 ^ { 4 }$

The following steps are applied to the number 123 in sequence to change the positions of its digits, and a three-digit number is obtained at each step.

1. In step 1, a number is obtained by switching the positions of the digits in the tens and hundreds places.
2. In step 2, a number is obtained by switching the positions of the digits in the ones and tens places of the number obtained in the previous step.

Continuing in this way, if the step number is odd, numbers are obtained by switching the positions of the digits in the tens and hundreds places of the number obtained in the previous step, and if the step number is even, by switching the positions of the digits in the ones and tens places of the number obtained in the previous step. Accordingly, which of the following is the number obtained after step 75?
A) 321
B) 312
C) 231
D) 213
E) 132