1. The equivariant middle term of $2 + \sqrt { 3 }$ and $2 - \sqrt { 3 }$ is
Answer: C
2. If $\left\{ a _ { n } \right\}$ is known to be an equivariant series, and if $a _ { 4 } = 15$ , then the sum of the first 7 terms of it is
Answer: D
3. The real numbers $m , 3,2$ are known to form an equidistant series, then the centroid of the conic curve $\frac { x ^ { 2 } } { m } + y ^ { 2 } = 1$ is
Answer: C
4. It is known that the sum of the first $n$ terms of the isomorphic series $\left\{ a _ { n } \right\}$ is $S _ { n } , ~ a _ { 1 } \neq 0 , ~ S _ { 8 } = 0$, then
Answer: C
5. In the isoperimetric series $\left\{ a _ { n } \right\}$, $a _ { 1 } = \frac { 9 } { 8 } , q = \frac { 2 } { 3 } , S _ { n } = \frac { 19 } { 8 }$, then $n =$
Answer: A
6. It is known that the common ratio of the isometric series $\left\{ a _ { n } \right\}$ in $a _ { 2 } = - 2 , a _ { 5 } = 16$ is
Answer: C
7. In the isomorphic series $\left\{ a _ { n } \right\}$, it is known that $a _ { 2 } + a _ { 6 } = 18$, then $a _ { 4 } = ($
Answer: A
8. The sum of the first $\left\{ a _ { n } \right\}$ terms of the isomorphic series $n$ is $S _ { n }$, and if $a _ { 4 } + a _ { 6 } = 12$, then the value of $S _ { 9 }$ is
Answer: C
9. $S n$ is known to be the sum of the first $n$ terms of the isomorphic series $\{ a n \}$ whose tolerance is not 0, $S _ { 9 } = 18$, $a m = 2$, and $m =$ ( ). INLINE_FORMULA_5]] ( )
Answer: B
10. The middle term of the equivalence of $\sqrt { 3 } - 1$ and $\sqrt { 3 } + 1$ is ( )
Answer: C
11. $\left\{ a _ { n } \right\}$ is an isometric series and $a _ { 7 } - 2 a _ { 4 } = - 1 , a _ { 3 } = 0$ , then the tolerance $d =$
Answer: B
12. In the isomorphic series $\{ a n \}$, $a _ { 3 } + a _ { 5 } = 10$, then $a _ { 1 } + a _ { 7 }$ is equal to ( )
Answer: C
13. It is known that the series $\left\{ a _ { n } \right\}$ in $a _ { 3 } = 2 , a _ { 7 } = 1$, and the series $\left\{ \frac { 1 } { a _ { n } + 1 } \right\}$ is an isometric series, then $a _ { 11 } =$
Answer: B
It is known that the first term $a _ { 1 } = 1$ of the positive isometric series $\left\{ a _ { n } \right\}$ and the sum of the first $n$ terms is $S _ { n }$, and $\mathrm { S } _ { 1 } , \mathrm {~S} _ { 2 } , S _ { 3 } - 2$ is an isotropic series, then $a _ { 4 } =$ is $a _ { 4 } =$. And $\mathrm { S } _ { 1 } , \mathrm {~S} _ { 2 } , S _ { 3 } - 2$ is an equal difference series, then $a _ { 4 } =$ ( ).
Answer: A
15.According to global skyscraper statistics, by 2019, Hefei City, Anhui Province, has 95 skyscrapers in Chinese cities ranked 10th, ranked 15th globally, and the tallest building under construction in the Evergrande Center in Hefei is designed in the shape of "bamboo joints", which not only embodies the power of extraordinary, but also symbolizes the strong will to grow upward, but also signals future prosperity and prosperity. The shape of the tallest building under construction of Evergrande Center in Hefei is designed in the form of "bamboo joints", which not only reflects the extraordinary strength, but also symbolizes the strong will of upward growth, and moreover, it foretells the future prosperity and prosperity. It and the inheritance of thousands of years of "micro-culture" complement each other, after the completion of the world's top ten skyscrapers, if the building consists of nine sections of "bamboo", the top part of the four sections of 228 meters high, the bottom part of the three sections of 204 meters high, and each section of the height of the changes in height evenly (i.e., the height of each section from top to bottom into an equal series), then the total height of the skyscraper is ( )
Answer: B
17. In $V A B C$, $a , ~ b , ~ c$ is the opposite side of angle $A , ~ B , ~ C$. If $a , ~ b , ~ c$ forms an equipartite series and $a ^ { 2 } - c ^ { 2 } = ( a - b ) c$, then the size of $A$ is
Answer: B
18. In the series $\left\{ a _ { n } \right\}$, $a _ { 1 } = 1 , a _ { 2 } = 2$, for $\forall n \in \mathbf { N } ^ { * } , a _ { n + 2 } = \frac { 5 } { 2 } a _ { n + 1 } - \frac { 3 } { 2 } a _ { n }$, then $a _ { 2021 } =$
Answer: C
19. The Chinese Remainder Theorem, also known as Sun Tzu's Theorem, is about division. The 2024 numbers from 1 to 2024 that are divisible by 3 and 1 and divisible by 5 are listed in descending order as $\left\{ a _ { n } \right\}$, preceded by $n _ { \text {项和为 } } S _ { n }$ and $S _ { 20 } - a _ { 10 } =$, and $S _ { 20 } - a _ { 10 } =$.
Answer: B
20. The standard logarithmic visual acuity chart (pictured) uses the "five-point recording method", which is a unique way of recording visual acuity in China. Standard logarithmic visual acuity table rows of square "$E$" word vision mark, and from the visual acuity of 5.1 vision mark in the rows upward, each line "$E$ is $\sqrt [ 10 ] { 10 }$ times the length of the side of "$E$" in the row below it, and if the length of the side of the reticle with visual acuity 4.0 is ${ } ^ { a }$, then the length of the side of the reticle with visual acuity 4.9 is | Standardized Log Distance Visual Acuity Chart | | | :--- | :--- | |  | | | [IMAGE_9]] | [IMAGE_9]] | [IMAGE_9]] | | | | | E ${ } ^ { 43 }$ | | | The Government of the United States of America has been working on a number of initiatives to improve the quality of education. | The following is a summary of the work done by the organization. | m Em $\boldsymbol { \Xi } 4.8$ | | | E m Yin $\omega$ E $\mathrm { m } \equiv$ 5.0 | | | | 5.0 | 5.1 | 5.0 | | 5.1 | | 5.2 | | |
Answer: D
21. Let the left and right foci of hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ be $F _ { 1 } , F _ { 2 }$, if there exists a point $P$ on the right branch of the hyperbola such that $\left| P F _ { 2 } \right| , \left| P F _ { 1 } \right| , \left| F _ { 1 } F _ { 2 } \right|$ is an equidistant series, then the range of values of the centrality of this hyperbola is ( )
Answer: A
The series $\left\{ a _ { n } \right\}$ is an isotropic series, if $\frac { a _ { 11 } } { a _ { 10 } } < - 1$, and the sum of the first $n$ terms of $S _ { n }$ has a maximum, then when $S _ { n }$ obtains When $S _ { n }$ gets the smallest positive value, $n =$
Answer: C
23. If the sum of the first $n$ terms of the series $\left\{ a _ { n } \right\}$ is $S _ { n }$ and satisfies $S _ { n } = \frac { 1 } { 2 } \left( a _ { n } + n - 1 \right)$, then the sum of the first 81 terms of the series $\left\{ n a _ { n } \right\}$ is ( )
Answer: A
24. In the isomorphic series $\left\{ a _ { n } \right\}$ with tolerance ${ } _ { d }$ and $S _ { 10 } = 4 S _ { 5 }$, then $\frac { a _ { 1 } } { d }$ is equal to
Answer: C
25. If $S _ { n }$ is known to be the sum of the first $n$ terms of the isomorphic series $\left\{ a _ { n } \right\}$ and $S _ { 3 } = 2 a _ { 1 }$, then the following conclusion is false
Answer: D
26. Read the block diagram on the right. If the input $n$ is 100, then the values of the output variables $S$ and $T$ are, in order, ( ) 
Answer: D
27. Given the following propositions, the number of correct propositions is ( ) (1) There exist two unequal real numbers $\alpha , \beta$ such that the equation $\sin ( \alpha + \beta ) = \sin \alpha + \sin \beta$ holds; (2) If the series $\left\{ a _ { n } \right\}$ is isometric and $m + n = s + t , m , n , s , t \in N ^ { * }$, then $a _ { m } + a _ { n } = a _ { s } + a _ { t }$; (3) If $S _ { n }$ is the sum of the first $n$ terms of the isoperimetric series $\left\{ a _ { n } \right\}$ and $S _ { n } = 3 \cdot 2 ^ { n } + A$, then $A = - 3$; (4) It is known that the sides opposite the three interior angles $A , B , C$ of $a , b , c$ are $a , b , c$ , and if $a ^ { 2 } + b ^ { 2 } > c ^ { 2 }$ , then $V A B C -$ is definitely an an acute triangle; ( )
Answer: C
28. It is known that the isoperimetric series $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } = 1 , a _ { 5 } = 4$, then $a _ { 2 } a _ { 3 } a _ { 4 } = ( )$
Answer: C
It is known that the series $\left\{ a _ { n } \right\}$ satisfies $a _ { n } = 3 \times 2 ^ { n - 1 } , n \in N ^ { * }$, which is now arranged in a serpentine array according to the following pattern (row $i$). INLINE_FORMULA_3]] and $i \in N ^ { * }$), and the ${ } ^ { j }$ number in the $i$ row from the left is ${ } ^ { ~ } { } _ { ( i , j ) } \left( i , j \in N ^ { * } \right.$ and $\left. { } ^ { j \leq i } \right)$ and $a _ { ( 21,21 ) } =$ is $a _ { n } = 3 \times 2 ^ { n - 1 } , n \in N ^ { * }$, and $a _ { n } = 3 \times 2 ^ { n - 1 } , n \in N ^ { * }$ is $i$. FORMULA_8]], then $a _ { ( 21,21 ) } =$ $a _ { 1 }$ $a _ { 2 } \quad a _ { 3 }$ $\begin{array} { l l l } a _ { 6 } & a _ { 5 } & a _ { 4 } \end{array}$ $\begin{array} { l l l l } a _ { 7 } & a _ { 8 } & a _ { 9 } & a _ { 10 } \end{array}$ $\begin{array} { l l l l l } a _ { 15 } & a _ { 14 } & a _ { 13 } & a _ { 12 } & a _ { 11 } \end{array}$ $\_\_\_\_$
Answer: B
30. The series $\left\{ a _ { n } \right\}$ is known to be an isoperimetric series, if $a _ { 5 } - a _ { 3 } = 12 , a _ { 6 } - a _ { 4 } = 24$, then $a _ { 2024 } =$
Answer: B
31. It is known that the sum of the first $n$ terms of the isomorphic series $\left\{ a _ { n } \right\}$ is $S _ { n }$ and $a _ { 1 } + a _ { 3 } + a _ { 8 } = 6$ is $S _ { 7 } =$.
Answer: D
32. If $\left\{ a _ { n } \right\}$ is known to be an isoperimetric series, and the middle term of the equality between $a _ { 2 } \cdot a _ { 3 } = 2 a _ { 1 } , a _ { 4 }$ and $2 a _ { 7 }$ is $\frac { 5 } { 4 }$, then $a _ { 5 } =$ is an isoperimetric series.
Answer: A
34. If all the terms of the isoperimetric series $\left\{ a _ { n } \right\}$ are positive, and $a _ { 5 } a _ { 6 } + a _ { 4 } a _ { 7 } = 6$, then $\log _ { 3 } \left( a _ { 1 } a _ { 2 } \cdots a _ { 10 } \right) =$
Answer: B
35. The sum of the first $n$ terms of the series $\left\{ a _ { n } \right\}$ is $S _ { n } , a _ { 1 } = 1 , a _ { n } = \left\{ \begin{array} { l } a _ { n - 1 } + 1 , n = 2 k \\ 2 a _ { n - 1 } + 1 , n = 2 k + 1 \end{array} \left( k \in \mathrm {~N} ^ { * } \right) \right.$. The following options are correct Ratio series
Answer: D
36. The positive isometric series $\left\{ a _ { n } \right\}$ is known to satisfy $2 a _ { 4 } + a _ { 3 } = a _ { 2 }$ , if there are two terms in the series $\left\{ a _ { n } \right\}$ and the equivariant middle term of $a _ { m } , a _ { n }$ is $\frac { a _ { 1 } } { 4 }$, then The minimum value of $\frac { 4 } { m } + \frac { 1 } { n }$ is ( )
Answer: A
37. It is known that the sum of the first $n$ terms of the isomorphic series $\left\{ a _ { n } \right\}$ is $n$ and $S _ { n }$ and $S _ { 25 } = 100$ is $a _ { 12 } + a _ { 14 } =$.
Answer: B
In the isoperimetric series $\left\{ a _ { n } \right\}$, $a _ { 1 } + a _ { 2 } + a _ { 3 } + a _ { 4 } = 20 , a _ { 5 } + a _ { 6 } + a _ { 7 } + a _ { 8 } = 10$ is known, and the sum of the first 16 terms of the series $\left\{ a _ { n } \right\}$ is $S _ { 16 }$.
Answer: B
39. If the series $\left\{ a _ { n } \right\}$ is an isometric series, $\left\{ b _ { n } \right\}$ is an isoperimetric series, and satisfies: $a _ { 1 } + a _ { 2019 } = \pi , b _ { 1 } \cdot b _ { 2019 } = 2$ , the function $f ( x ) = \sin x$, then $f \left( \frac { a _ { 1009 } + a _ { 1011 } } { 1 + b _ { 1009 } b _ { 1011 } } \right) =$
Answer: C
40. The series $\left\{ a _ { n } \right\}$ satisfies $a _ { n + 1 } = \left( 2 \left| \sin \frac { n \pi } { 2 } \right| - 1 \right) a _ { n } + n , n \in \mathrm {~N} ^ { * }$, then the sum of the first 20 terms of the series $\left\{ a _ { n } \right\}$ is High School Mathematics Assignment, October 29, 2025
Answer: B
数列
38 Practice Questions
Practice with Chinese questions to prepare for the CSCA exam. You can toggle translations while practicing.
A series is a sequence of numbers in a certain order in mathematics, mainly including equal difference series and equal ratio series. In the CSCA exam, series questions often involve the general formula, the calculation of the sum of the first n terms, and the application of series properties in geometric and algebraic problems. Mastering the basic formulas and applying them flexibly is the key to solving the questions.
It is recommended to familiarize yourself with the core formulas of the equal-difference and equal-ratio series, and to master the property of the middle term and the solution of cross-point synthesis problems through practice problems.
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