Let $S _ { a }$ denote the sum of first $n$ terms an arithmetic progression. If $S _ { 20 } = 790$ and $S _ { 10 } = 145$, then $S _ { 15 } - S _ { 5 }$ is (1) 395 (2) 390 (3) 405 (4) 410
A software company sets up $m$ number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of $m$ is equal to: (1) 150 (2) 180 (3) 160 (4) 125
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
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
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
Q87. If a function $f$ satisfies $f ( \mathrm {~m} + \mathrm { n } ) = f ( \mathrm {~m} ) + f ( \mathrm { n } )$ for all $\mathrm { m } , \mathrm { n } \in \mathbf { N }$ and $f ( 1 ) = 1$, then the largest natural number $\lambda$ such that $\sum _ { k = 1 } ^ { 2022 } f ( \lambda + k ) \leq ( 2022 ) ^ { 2 }$ is equal to $\_\_\_\_$ Q88. Let $f : ( 0 , \pi ) \rightarrow \mathbf { R }$ be a function given by $f ( x ) = \left\{ \begin{array} { c c } \left( \frac { 8 } { 7 } \right) ^ { \frac { \tan 8 x } { \tan 7 x } } , & 0 < x < \frac { \pi } { 2 } \\ \mathrm { a } - 8 , & x = \frac { \pi } { 2 } \\ ( 1 + | \cot x | ) ^ { \mathrm { b } } | \tan x | , & \frac { \pi } { 2 } < x < \pi \end{array} \right.$ where $\mathrm { a } , \mathrm { b } \in \mathbf { Z }$. If $f$ is continuous at $x = \frac { \pi } { 2 }$, then $\mathrm { a } ^ { 2 } + \mathrm { b } ^ { 2 }$ is equal to
1. A sequence $a_{1} + 2, \cdots, a_{k} + 2k, \cdots, a_{10} + 20$ has ten terms, and their sum is 240. Then the value of $a_{1} + \cdots + a_{k} + \cdots + a_{10}$ is (1) 31 (2) 120 (3) 130 (4) 185 (5) 218
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
The sequence $a _ { n }$ is defined by the rule: $$a _ { n } = ( - 1 ) ^ { n } - ( - 1 ) ^ { n - 1 } + ( - 1 ) ^ { n + 2 } \text { for } n \geq 1$$ Find the value of $$\sum _ { n = 1 } ^ { 39 } a _ { n }$$ A - 39 B - 3 C - 1 D 0 E 1 F 3 G 39
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 function f on the set of real numbers is defined for every real number x as $$f ( x ) = \left\{ \begin{array} { c c }
x + 2 , & x < 0 \\
x , & x \geq 0
\end{array} \right.$$ Accordingly, what is the value of the sum $\sum _ { k = - 3 } ^ { 4 } f ( k )$? A) 8 B) 10 C) 12 D) 14 E) 16
Filiz creates cup towers by placing identical cardboard cups inside each other. The distance between the bases of every two consecutive cups is equal in all the cup towers she creates. Then, she places these towers on a table and measures their heights. Filiz observes that the sum of the heights of two towers with 6 and 9 cups equals the height of the tower with 18 cups. Accordingly, to what height of a cup tower is the sum of the heights of two towers with 8 and 12 cups equal? A) 23 B) 24 C) 26 D) 27 E) 29
The table below shows some musical note symbols and the duration lengths of these note symbols. Accordingly, what is the sum of the duration lengths of the musical note symbols given above? A) $\frac{3}{2}$ B) $\frac{7}{4}$ C) $\frac{5}{4}$ D) $\frac{13}{8}$ E) $\frac{15}{8}$