If $\frac { 1 ^ { 3 } + 2 ^ { 3 } + 3 ^ { 3 } \ldots\ldots \text { upto } n \text { terms } } { 1 \cdot 3 + 2 \cdot 5 + 3 \cdot 7 + \ldots.. \text { upto } n \text { terms } } = \frac { 9 } { 5 }$ then the value of $n$ is

If $a_n = \frac{-2}{4n^2 - 16n + 15}$, then $a_1 + a_2 + \ldots + a_{25}$ is equal to:
(1) $\frac{51}{144}$
(2) $\frac{49}{138}$
(3) $\frac{50}{141}$
(4) $\frac{52}{147}$

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

If $S _ { n } = 4 + 11 + 21 + 34 + 50 + \ldots$ to $n$ terms, then $\frac { 1 } { 60 } \left( S _ { 29 } - S _ { 9 } \right)$ is equal to
(1) 223
(2) 226
(3) 220
(4) 227

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) $- \frac { 55 } { 109 }$

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) $\frac { 1 } { 3 }$

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

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) $7 . \overline { 9 }$

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