If $8 = 3 + \frac { 1 } { 4 } ( 3 + p ) + \frac { 1 } { 4 ^ { 2 } } ( 3 + 2 p ) + \frac { 1 } { 4 ^ { 3 } } ( 3 + 3 p ) + \ldots \infty$, then the value of $p$ is

Let the positive integers be written in the form : If the $k ^ { \text {th} }$ row contains exactly $k$ numbers for every natural number $k$, then the row in which the number 5310 will be, is $\_\_\_\_$

An arithmetic progression is written in the following way

			2
	11	5	14	8	17
20		23		26		29

The sum of all the terms of the $10 ^ { \text {th} }$ row is $\_\_\_\_$

If $\left( \frac { 1 } { \alpha + 1 } + \frac { 1 } { \alpha + 2 } + \ldots \ldots + \frac { 1 } { \alpha + 1012 } \right) - \left( \frac { 1 } { 2 \cdot 1 } + \frac { 1 } { 4 \cdot 3 } + \frac { 1 } { 6 \cdot 5 } + \ldots . + \frac { 1 } { 2024 \cdot 2023 } \right) = \frac { 1 } { 2024 }$, then $\alpha$ is equal to $\_\_\_\_$

If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to
(1) - 1080
(2) - 1020
(3) - 1200
(4) - 120

Consider an A.P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its $11^{\text{th}}$ term is:
(1) 90
(2) 84
(3) 122
(4) 108

Let $S_n = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \ldots$ upto $n$ terms. If the sum of the first six terms of an A.P. with first term $-p$ and common difference $p$ is $\sqrt{2026 S_{2025}}$, then the absolute difference between $20^{\text{th}}$ and $15^{\text{th}}$ terms of the A.P. is
(1) 20
(2) 90
(3) 45
(4) 25

Suppose that the number of terms in an A.P. is $2k , k \in N$. If the sum of all odd terms of the A.P. is 40 , the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27 , then k is equal to :
(1) 6
(2) 5
(3) 8
(4) 4

Let $\mathrm { T } _ { \mathrm { r } }$ be the $\mathrm { r } ^ { \text {th} }$ term of an A.P. If for some $\mathrm { m } , T _ { m } = \frac { 1 } { 25 } , T _ { 25 } = \frac { 1 } { 20 }$, and $20 \sum _ { \mathrm { r } = 1 } ^ { 25 } T _ { \mathrm { r } } = 13$, then $5 \mathrm { m } \sum _ { \mathrm { r } = \mathrm { m } } ^ { 2 \mathrm { m } } T _ { \mathrm { r } }$ is equal to
(1) 98
(2) 126
(3) 142
(4) 112

In an arithmetic progression, if $S_{40} = 1030$ and $S_{12} = 57$, then $S_{30} - S_{10}$ is equal to:
(1) 525
(2) 510
(3) 515
(4) 505

The roots of the quadratic equation $3 x ^ { 2 } - \mathrm { p } x + \mathrm { q } = 0$ are $10 ^ { \text {th} }$ and $11 ^ { \text {th} }$ terms of an arithmetic progression with common difference $\frac { 3 } { 2 }$. If the sum of the first 11 terms of this arithmetic progression is 88, then $q - 2 p$ is equal to

Let $a _ { 1 } , a _ { 2 } , \ldots , a _ { 2024 }$ be an Arithmetic Progression such that $a _ { 1 } + \left( a _ { 5 } + a _ { 10 } + a _ { 15 } + \ldots + a _ { 2020 } \right) + a _ { 2024 } = 2233$. Then $a _ { 1 } + a _ { 2 } + a _ { 3 } + \ldots + a _ { 2024 }$ is equal to $\_\_\_\_$

The interior angles of a polygon with $n$ sides, are in an A.P. with common difference $6 ^ { \circ }$. If the largest interior angle of the polygon is $219 ^ { \circ }$, then n is equal to

Consider a sequence $\{a_n\}$ $(n = 1, 2, 3, \cdots)$ where the sum of the first $n$ terms is
$$\sum_{k=1}^{n} a_k = n^2 + 3n$$
(1) Then $a_n = \mathbf{A}\, n + \mathbf{B}$.
(2) For the sequence $\{b_n\}$ $(n = 1, 2, 3, \cdots)$, where $b_n = n^2 - 5n - 6$, the number of terms satisfying $b_n < 0$ is $\mathbf{C}$, and the sum of such terms is $-\mathbf{DE}$.
(3) It follows that for the sequences $\{a_n\}$ and $\{b_n\}$ in (1) and (2),
$$\sum_{k=1}^{n} \frac{k^2 b_k}{a_k} = \frac{1}{\mathbf{F}}\, n(n + \mathbf{G})\left(n^2 - \mathbf{H}\, n - \mathbf{I}\right).$$

Let $\{a_n\}$ be a sequence such that the sum $S_n$ of the terms from the first term to the $n$-th term is $$S_n = \frac{n^2 - 17n}{4},$$ and let $\{b_n\}$ be the sequence defined by $$b_n = a_n \cdot a_{n+5} \quad (n = 1, 2, 3, \cdots)$$
(1) For $\mathbf{A}$ $\sim$ $\mathbf{C}$ in the following sentences, choose the correct answer from among choices (0) $\sim$ (9) below.
Let us find the sum $T_n$ of the terms of sequence $\{b_n\}$ from the first term to the $n$-th term.
Since $a_n = \mathbf{A}$, we have $b_n = \mathbf{B}$. Hence we obtain $$T_n = \mathbf{C}.$$
(0) $\frac{n-7}{2}$
(1) $\frac{n-9}{2}$
(2) $\frac{n-11}{2}$
(3) $\frac{n^2 - 12n + 27}{4}$
(4) $\frac{n^2 - 13n + 36}{4}$
(5) $\frac{n^2 - 14n + 45}{4}$ (6) $\frac{n(n^2 - 17n + 83)}{12}$ (7) $\frac{n(n^2 - 17n + 89)}{12}$ (8) $\frac{n(n^2 - 18n + 83)}{12}$ (9) $\frac{n(n^2 - 18n + 89)}{12}$
(2) Next, let us find the minimum value of $T_n$.
When $n \leqq \mathbf{D}$ or $\mathbf{EF} \leqq n$, we see that $b_n > 0$. On the other hand, when $\mathbf{G} \leqq n \leqq \mathbf{H}$, we see that $b_n < 0$.
Hence $T_n$ is minimized at $n = \mathbf{I}$, $n = \mathbf{J}$ and $n = \mathbf{K}$, and its minimum value is $\mathbf{L}$. (Answer in the order such that $\mathbf{I} < \mathbf{J} < \mathbf{K}$.)

Let the arithmetic sequence $\left\langle a _ { n } \right\rangle$ have first term $a _ { 1 }$ and common difference $d$ both positive, and $\log a _ { 1 } , \log a _ { 3 } , \log a _ { 6 }$ also form an arithmetic sequence in order. Select the common difference of the sequence $\log a _ { 1 } , \log a _ { 3 } , \log a _ { 6 }$.
(1) $\log d$
(2) $\log \frac { 2 } { 3 }$
(3) $\log \frac { 3 } { 2 }$
(4) $\log 2 d$
(5) $\log 3 d$

A company has two new employees, A and B, who start at the same time with the same starting salary. The company promises the following salary adjustment methods for employees A and B:
Employee A: After 3 months of work, starting the next month, monthly salary increases by 200 yuan; thereafter, salary is adjusted in the same manner every 3 months.
Employee B: After 12 months of work, starting the next month, monthly salary increases by 1000 yuan; thereafter, salary is adjusted in the same manner every 12 months.
Based on the above description, select the correct options.
(1) After 8 months of work, the monthly salary in the 9th month is 600 yuan more than in the 1st month
(2) After one year of work, in the 13th month, employee A's monthly salary is higher than employee B's
(3) After 18 months of work, in the 19th month, employee A's monthly salary is higher than employee B's
(4) After 18 months of work, the total salary received by employee A is less than the total salary received by employee B
(5) After two years of work, in the 12 months of the 3rd year, there are exactly 3 months where employee A's monthly salary is higher than employee B's

An electronic billboard continuously alternates between playing advertisements A and B ($A$, $B$, $A$, $B \ldots$), with each advertisement playing for $T$ minutes (where $T$ is an integer). A person passes by just as advertisement A starts playing. 30 minutes later, the person returns to the location and sees advertisement B just starting to play. Select the options that could be the value of $T$.
(1) $15$ (2) $10$ (3) $8$ (4) $6$ (5) $5$

From the 20 integers from 1 to 20, select three distinct numbers $a$, $b$, $c$ that form an arithmetic sequence with $a < b < c$. The number of ways to choose $(a, b, c)$ is $\square\square$.

An arithmetic sequence has a first term of 1, a last term of 81, and 9 is also in the sequence. Let the number of terms in this sequence be $n$, where $n \leq 100$ . Select the correct options.
(1) $n$ is odd
(2) 41 must be in this arithmetic sequence
(3) The common difference of all arithmetic sequences satisfying the conditions is an integer
(4) There are 10 arithmetic sequences satisfying the conditions
(5) If $n$ is a multiple of 7, then $n = 21$

The sequences $\{a_{n}\}$ and $\{b_{n}\}$ are defined as follows. $$a_{n} = \begin{cases} 0, & \text{if } n \equiv 0 \pmod{3} \\ n, & \text{if } n \equiv 1 \pmod{3} \\ -n, & \text{if } n \equiv 2 \pmod{3} \end{cases}, \quad b_{n} = \sum_{k=0}^{n} a_{k}$$ Accordingly, what is $b_{4}$?
A) $-2$
B) $-1$
C) $0$
D) $2$
E) $3$

On day 1, Ismail puts one of each of the following coins into his piggy bank: 5 Kr, 10 Kr, 25 Kr, 50 Kr, and 1 TL. On day 2, he puts two of each, and continuing in this manner, on day n he puts n of each.
If Ismail has saved 104.5 TL in his piggy bank, what is n?
A) 10 B) 11 C) 12 D) 13 E) 14

The sum of all two-digit natural numbers with digit A in the units place is 504. What is A?
A) 5
B) 6
C) 7
D) 8
E) 9

$$\frac { [ ( n + 1 ) ! ] ^ { 2 } + ( n ! ) ^ { 2 } } { [ ( n + 1 ) ! ] ^ { 2 } - ( n ! ) ^ { 2 } } = \frac { 61 } { 60 }$$
Given this, what is n?
A) 9
B) 10
C) 12
D) 13
E) 15

For integers $\mathrm { x } , \mathrm { y }$ and z
$$2 x = 3 y = 5 z$$
Given this, which of the possible values of the sum $x + y + z$ is closest to 100?
A) 93
B) 96
C) 98
D) 103
E) 105