If $x _ { 1 } , x _ { 2 } , \ldots\ldots , x _ { n }$ and $\frac { 1 } { h _ { 1 } } , \frac { 1 } { h _ { 2 } } , \ldots\ldots , \frac { 1 } { h _ { n } }$ are two A.P.s such that $x _ { 3 } = h _ { 2 } = 8 \& x _ { 8 } = h _ { 7 } = 20$, then $x _ { 5 } \cdot h _ { 10 }$ is equal to
(1) 3200
(2) 1600
(3) 2650
(4) 2560

If $x _ { 1 } , x _ { 2 } , \ldots , x _ { n }$ and $\frac { 1 } { h _ { 1 } } , \frac { 1 } { h ^ { 2 } } , \ldots \ldots \frac { 1 } { h _ { n } }$ are two A.P's such that $x _ { 3 } = h _ { 2 } = 8$ and $x _ { 8 } = h _ { 7 } = 20$, then $x _ { 5 }$. $h _ { 10 }$ equals.
(1) 2560
(2) 2650
(3) 3200
(4) 1600

If the lengths of the sides of a triangle are in A.P and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is:
(1) $3 : 4 : 5$
(2) $5 : 6 : 7$
(3) $5 : 9 : 13$
(4) $4 : 5 : 6$

If the sum and product of the first three terms in an A.P. are 33 and 1155, respectively, then a value of its $11^{\text{th}}$ term is:
(1) $- 25$
(2) $- 35$
(3) $25$
(4) $- 36$

Five numbers are in A.P., whose sum is 25 and product is 2520. If one of these five numbers is $- \frac { 1 } { 2 }$, then the greatest number amongst them is
(1) 27
(2) 7
(3) $\frac { 21 } { 2 }$
(4) 16

Let $a _ { 1 } , a _ { 2 } , \ldots , a _ { n }$ be a given A.P. whose common difference is an integer and $S _ { n } = a _ { 1 } + a _ { 2 } + \ldots + a _ { n }$. If $a _ { 1 } = 1 , a _ { n } = 300$ and $15 \leq n \leq 50$, then the ordered pair $\left( \mathrm { S } _ { n - 4 } , a _ { n - 4 } \right)$ is equal to:
(1) $( 2490,249 )$
(2) $( 2480,249 )$
(3) $( 2480,248 )$
(4) $( 2490,248 )$

If $3 ^ { 2 \sin 2 \alpha - 1 } , 14$ and $3 ^ { 4 - 2 \sin 2 \alpha }$ are the first three terms of an A.P. for some $\alpha$, then the sixth term of this A.P. is
(1) 66
(2) 81
(3) 65
(4) 78

Let $a , b , c$ be in arithmetic progression. Let the centroid of the triangle with vertices $( a , c ) , ( 2 , b )$ and $( a , b )$ be $\left( \frac { 10 } { 3 } , \frac { 7 } { 3 } \right)$. If $\alpha , \beta$ are the roots of the equation $a x ^ { 2 } + b x + 1 = 0$, then the value of $\alpha ^ { 2 } + \beta ^ { 2 } - \alpha \beta$ is:
(1) $- \frac { 71 } { 256 }$
(2) $\frac { 69 } { 256 }$
(3) $\frac { 71 } { 256 }$
(4) $- \frac { 69 } { 256 }$

If $a _ { 1 } , a _ { 2 } , a _ { 3 } \ldots$ and $b _ { 1 } , b _ { 2 } , b _ { 3 } \ldots$ are A.P. and $a _ { 1 } = 2 , a _ { 10 } = 3 , a _ { 1 } b _ { 1 } = 1 = a _ { 10 } b _ { 10 }$ then $a _ { 4 } b _ { 4 }$ is equal to
(1) $\frac { 28 } { 27 }$
(2) $\frac { 28 } { 24 }$
(3) $\frac { 23 } { 26 }$
(4) $\frac { 22 } { 23 }$

Let $a_1 = 8, a_2, a_3, \ldots, a_n$ be an A.P. If the sum of its first four terms is 50 and the sum of its last four terms is 170, then the product of its middle two terms is $\_\_\_\_$.

Let $f ( x ) = 2 x ^ { n } + \lambda , \lambda \in \mathbb { R } , \mathrm { n } \in \mathbb { N }$, and $f ( 4 ) = 133 , f ( 5 ) = 255$. Then the sum of all the positive integer divisors of $( f ( 3 ) - f ( 2 ) )$ is
(1) 61
(2) 60
(3) 58
(4) 59

The $20 ^ { \text {th} }$ term from the end of the progression $20,19 \frac { 1 } { 4 } , 18 \frac { 1 } { 2 } , 17 \frac { 3 } { 4 } , \ldots , - 129 \frac { 1 } { 4 }$ is :-
(1) - 118
(2) - 110
(3) - 115
(4) - 100

For $x \geqslant 0$, the least value of K , for which $4 ^ { 1 + x } + 4 ^ { 1 - x } , \frac { \mathrm {~K} } { 2 } , 16 ^ { x } + 16 ^ { - x }$ are three consecutive terms of an A.P., is equal to :
(1) 8
(2) 4
(3) 10
(4) 16

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

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

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

The following are known about a set A.

• It consists of 6 consecutive odd natural numbers.
• The sum of the elements in the set equals 4 times the largest element.

Accordingly, what is the largest element of set $A$?
A) 21
B) 19
C) 17
D) 15
E) 13

The following information is known about the athletes participating in a race.

• The jersey numbers of male athletes are consecutive odd numbers starting from 1.
• The jersey numbers of female athletes are consecutive even numbers starting from 2.
• The number of male athletes is 3 times the number of female athletes.
• The largest jersey number given to male athletes is 83.

Given this, what is the largest jersey number given to female athletes?
A) 28
B) 30
C) 32
D) 34
E) 36

The following information is known about the monthly salaries of Ahmet and Beyza, who started work on the same day at a workplace.

• Ahmet's initial salary is 2500 TL.
• Ahmet's salary increases by 50 TL every 4 months.
• Beyza's salary increases by 100 TL every 6 months.

Given that their salaries are equal 6 years after they receive their first salaries, what is Beyza's initial salary in TL?
A) 2000
B) 2100
C) 2200
D) 2300
E) 2400

Let $\left( a _ { n } \right)$ be an arithmetic sequence such that $$\begin{aligned}& a _ { 10 } + a _ { 7 } = 6 \\& a _ { 9 } - a _ { 6 } = 1\end{aligned}$$ the following equalities are given.\ Accordingly, what is $a _ { 1 }$?\ A) $\frac { 7 } { 3 }$\ B) $\frac { 5 } { 2 }$\ C) $\frac { 4 } { 3 }$\ D) $\frac { 5 } { 6 }$\ E) $\frac { 1 } { 2 }$

For an arithmetic sequence $(a_n)$ with distinct terms and common difference $r$,
$$\begin{aligned} & a _ { 1 } = 3 \cdot r \\ & a _ { 6 } = a _ { 2 } \cdot a _ { 4 } \end{aligned}$$
the equalities are given.
Accordingly, what is $\mathbf { a } _ { \mathbf { 1 0 } }$?
A) 10 B) 8 C) 6 D) 4 E) 2

Ceyda plans to take an equal number of steps each day for a week. The graph below shows the difference between the number of steps Ceyda took daily and the number of steps she planned to take during this week.
For example, Ceyda took 50 more steps than planned on Monday and 100 fewer steps than planned on Tuesday.
On Friday, Ceyda took 165 more steps than on Thursday and 10 fewer steps than on Saturday, and after 7 days, the total number of steps she took was equal to the number of steps she initially planned to take.
Accordingly, how many more steps did Ceyda take on Friday than the number of steps she planned to take daily?
A) 85
B) 90
C) 95
D) 100
E) 105

For an arithmetic sequence $(a_n)$:
$$\begin{gathered} a _ { 2 } = 2 a _ { 1 } + 1 \\ a _ { 6 } + a _ { 22 } = 34 \end{gathered}$$
equalities are given.
Accordingly, what is $a _ { 7 }$?
A) $61^3$
B) 7
C) 8
D) 9
E) 10

A flower bed with a height of 1.2 meters has five shelves at equal intervals in its left compartment and six shelves at equal intervals in its right compartment. The shelves at the bottom and top of these two compartments are at equal heights. A flower has been placed on the 4th shelf in the left compartment and on the 3rd shelf in the right compartment of the flower bed as shown in the figure.
Accordingly, what is the sum of the heights from the ground of the shelves where the flowers are located in meters?
A) 1.38 B) 1.36 C) 1.34 D) 1.32 E) 1.30

Two balloons are hung on a string stretched between two walls as shown in the figure. Between these two balloons, 2 white balloons or 4 yellow balloons are to be hung such that the distance between the points where any two adjacent balloons are attached to the string is equal.
The distance between the points where any two adjacent balloons are attached to the string is 18 cm more when white balloons are hung compared to when yellow balloons are hung.
Accordingly, what is the distance in cm between the points where the two initially hung balloons are attached to the string?
A) 135 B) 144 C) 153 D) 162 E) 171