Q82. Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ be in an arithmetic progression of positive terms. Let $\mathrm { A } _ { \mathrm { k } } = \mathrm { a } _ { 1 } ^ { 2 } - \mathrm { a } _ { 2 } ^ { 2 } + \mathrm { a } _ { 3 } ^ { 2 } - \mathrm { a } _ { 4 } ^ { 2 } + \ldots + \mathrm { a } _ { 2 \mathrm { k } - 1 } ^ { 2 } - \mathrm { a } _ { 2 \mathrm { k } } ^ { 2 }$. If $\mathrm { A } _ { 3 } = - 153 , \mathrm {~A} _ { 5 } = - 435$ and $\mathrm { a } _ { 1 } ^ { 2 } + \mathrm { a } _ { 2 } ^ { 2 } + \mathrm { a } _ { 3 } ^ { 2 } = 66$, then $\mathrm { a } _ { 17 } - \mathrm { A } _ { 7 }$ is equal to $\_\_\_\_$

Q82. An arithmetic progression is written in the following way

			2
	11	5	14	8	17
20	math	23		26		29

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

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

If $a + b + c = 1$ and $a < b < c , a , b , c \in R$ and $a ^ { \mathbf { 2 } } , 2 b ^ { \mathbf { 2 } } , c ^ { \mathbf { 2 } }$ are in G.P. and $a , b , c$ are in A.P. then find the value of $9 \left( a ^ { 2 } + b ^ { 2 } + c ^ { 2 } \right) =$ ?

$\frac { 6 } { 3 ^ { 26 } } + \frac { 10 } { 3 ^ { 25 } } + \frac { 10.2 } { 3 ^ { 24 } } + \frac { 10.2 ^ { 2 } } { 3 ^ { 23 } } + \cdots + \frac { 10.2 ^ { 24 } } { 3 }$ is equal to

If sum of first 4 terms of an A.P is 6 and sum of first 6 terms is 4, then sum of first 12 terms of an A.P is
(A) -21
(B) -22
(C) -23
(D) - 24

Consider an A.P $a _ { 1 } , a _ { 2 } \cdots a _ { n } ; a _ { 1 } > 0 , a _ { 2 } - a _ { 1 } = \frac { - 3 } { 4 } , a _ { n } = \frac { 1 } { 4 } a _ { 1 }$ and $\sum _ { i = 1 } ^ { n } a _ { i } = \frac { 525 } { 2 }$ then $\sum _ { i = 1 } { 17 } a _ { i }$ is equal to
(A) 189
(B) 238
(C) 276
(D) 258

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 the sequence $\left\{ a _ { n } \right\} ( n = 1,2,3 , \cdots )$ be an arithmetic progression satisfying
$$a _ { 2 } = 2 , \quad a _ { 6 } = 3 a _ { 3 } .$$
Then, consider the series $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } } { r ^ { a _ { n } } }$, where $r$ is a positive real number.
(1) When we denote the first term of $\left\{ a _ { n } \right\}$ by $a$, and the common difference by $d$, we have
$$a = \mathbf { A B } , \quad d = \mathbf { C } .$$
(2) The series $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } } { r ^ { a _ { n } } }$ is an infinite geometric series where the first term is $\square r^{\mathbf{E}}$, and the common ratio is $\dfrac { \mathbf { F } } { r^{\mathbf{G}} }$. Hence, this series converges when
$$r > 3 ^ { \frac { \mathbf { H } } { \mathbf{I} } } ,$$
and its sum $S$ is
$$S = \frac { \square \mathbf { J } \, r ^ { \mathbf { K } } } { r ^ { \mathbf { L } } - \mathbf { M } } .$$
(3) This sum $S$ is minimized at
$$r = \mathbf { N } ^ { \frac { \mathbf { O } } { 2 } } .$$

There is a staircase of 10 steps which we are to climb. We can go up one step at a time or two steps at a time, but we have to use each method at least once.
(1) Suppose we can go up two steps at a time twice or more in a row. (i) If we climb the staircase going up two steps at a time just 3 times, we will go up one step at a time just $\mathbf{P}$ times, and there are $\mathbf{QR}$ different ways of climbing the staircase. (ii) If we can go up two steps at a time twice or more in a row, there are altogether $\mathbf{ST}$ different ways of climbing the staircase.
(2) Suppose we cannot go up two steps at a time twice or more in a row. (i) If we climb the staircase going up two steps at a time just twice, we will go up one step at a time just $\mathbf{U}$ times, and there are $\mathbf{VW}$ different ways of climbing the staircase. (ii) If we cannot go up two steps at a time twice or more in a row, there are altogether $\mathbf{XY}$ different ways of climbing the staircase.

8. Suppose real numbers $a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 }$ form an arithmetic sequence, and satisfy $0 < a _ { 1 } < 2$ and $a _ { 3 } = 4$. If $b _ { n } = 2 ^ { a _ { n } }$ is defined, which of the following options are correct?
(1) $b _ { 1 } , b _ { 2 } , b _ { 3 } , b _ { 4 }$ form a geometric sequence.
(2) $b _ { 1 } < b _ { 2 }$.
(3) $b _ { 2 } > 4$.
(4) $b _ { 4 } > 32$.
(5) $b _ { 2 } \times b _ { 4 } = 256$.

8. Let $a _ { 1 } , a _ { 2 } , a _ { 3 }$ form an arithmetic sequence, and $b _ { 1 } , b _ { 2 } , b _ { 3 }$ form a geometric sequence, where all six numbers are real. Which of the following statements are correct?
(1) It is possible for both $a _ { 1 } < a _ { 2 }$ and $a _ { 2 } > a _ { 3 }$ to hold simultaneously
(2) It is possible for both $b _ { 1 } < b _ { 2 }$ and $b _ { 2 } > b _ { 3 }$ to hold simultaneously
(3) If $a _ { 1 } + a _ { 2 } < 0$, then $a _ { 2 } + a _ { 3 } < 0$
(4) If $b _ { 1 } b _ { 2 } < 0$, then $b _ { 2 } b _ { 3 } < 0$
(5) If $b _ { 1 } , b _ { 2 } , b _ { 3 }$ are all positive integers and $b _ { 1 } < b _ { 2 }$, then $b _ { 1 }$ divides $b _ { 2 }$

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 robot cat starts from the origin on a number line and moves in the positive direction. Its movement method is as follows: With an 8-second cycle, it moves at a constant speed of 4 units per second for 6 seconds, then rests for 2 seconds. Continuing this way, the robot cat will reach the position with coordinate 116 on the number line after $\underbrace{(14)(15)}$ seconds from the start of movement.

A sequence of five real numbers $a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 } , a _ { 5 }$ where each term is greater than 1, and between any two adjacent terms, one is twice the other. If $a _ { 1 } = \log _ { 10 } 36$, how many possible values can $a _ { 5 }$ have?
(1) 3
(2) 4
(3) 5
(4) 7
(5) 8

A light show display uses color-changing flashing lights. After each activation, the flashing color changes periodically according to the following sequence: Blue–White–Red–White–Blue–White–Red–White–Blue–White–Red–White…, with one cycle every four flashes. Blue light lasts 5 seconds each time, white light lasts 2 seconds each time, and red light lasts 6 seconds each time. Assuming the time to change lights is negligible, select the light color(s) between the 99th and 101st seconds after activation.
(1) All blue lights
(2) All white lights
(3) All red lights
(4) Blue light first, then white light
(5) White light first, then red light

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

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 $\square\square$.

Let $a \in \{-6, -4, -2, 2, 4, 6\}$ be the leading coefficient of a real-coefficient cubic polynomial $f(x)$. If the graph of the function $y = f(x)$ intersects the $x$-axis at three points whose $x$-coordinates form an arithmetic sequence with first term $-7$ and common difference $a$, how many values of $a$ satisfy $f(0) > 0$?
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Let the real numbers $a_{1}, a_{2}, \ldots, a_{9}$ form an arithmetic sequence with common difference 2, where $a_{1} \neq 0$ and $a_{3} > 0$. If $\log_{2} a_{3}$, $\log_{2} b$, $\log_{2} a_{9}$ form an arithmetic sequence in order, where $b$ is one of $a_{4}, a_{5}, a_{6}, a_{7}, a_{8}$, then $a_{9} = $ . (Express as a simplified fraction)