LFM Pure and Mechanics

csat-suneung 2019 Q29 4 marks Arithmetic-Geometric Hybrid Problem View

An arithmetic sequence $\left\{ a _ { n } \right\}$ with first term a natural number and common difference a negative integer, and a geometric sequence $\left\{ b _ { n } \right\}$ with first term a natural number and common ratio a negative integer, satisfy the following conditions. Find the value of $a _ { 7 } + b _ { 7 }$. [4 points] (가) $\sum _ { n = 1 } ^ { 5 } \left( a _ { n } + b _ { n } \right) = 27$ (나) $\sum _ { n = 1 } ^ { 5 } \left( a _ { n } + \left| b _ { n } \right| \right) = 67$ (다) $\sum _ { n = 1 } ^ { 5 } \left( \left| a _ { n } \right| + \left| b _ { n } \right| \right) = 81$

csat-suneung 2020 Q15 4 marks Optimization Involving an Arithmetic Sequence View

For an arithmetic sequence with first term 50 and common difference $- 4$, let $S _ { n }$ denote the sum of the first $n$ terms. What is the value of the natural number $m$ that maximizes $\sum _ { k = m } ^ { m + 4 } S _ { k }$? [4 points]
(1) 8
(2) 9
(3) 10
(4) 11
(5) 12

csat-suneung 2021 Q25 3 marks Summation of Derived Sequence from AP View

For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 3, if $\sum _ { k = 1 } ^ { 5 } a _ { k } = 55$, find the value of $\sum _ { k = 1 } ^ { 5 } k \left( a _ { k } - 3 \right)$. [3 points]

csat-suneung 2022 Q2 3 marks Find Specific Term from Given Conditions View

For an arithmetic sequence $\left\{ a _ { n } \right\}$, $$a _ { 2 } = 6 , \quad a _ { 4 } + a _ { 6 } = 36$$ what is the value of $a _ { 10 }$? [3 points]
(1) 30
(2) 32
(3) 34
(4) 36
(5) 38

csat-suneung 2023 Q7 3 marks Telescoping or Non-Standard Summation Involving an AP View

An arithmetic sequence $\left\{ a _ { n } \right\}$ with all positive terms and equal first term and common difference satisfies $$\sum _ { k = 1 } ^ { 15 } \frac { 1 } { \sqrt { a _ { k } } + \sqrt { a _ { k + 1 } } } = 2$$ What is the value of $a _ { 4 }$? [3 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

csat-suneung 2023 Q18 3 marks Compute Partial Sum of an Arithmetic Sequence View

For two sequences $\left\{ a _ { n } \right\}$ and $\left\{ b _ { n } \right\}$, $$\sum _ { k = 1 } ^ { 5 } \left( 3 a _ { k } + 5 \right) = 55 , \quad \sum _ { k = 1 } ^ { 5 } \left( a _ { k } + b _ { k } \right) = 32$$ What is the value of $\sum _ { k = 1 } ^ { 5 } b _ { k }$? [3 points]

csat-suneung 2024 Q11 4 marks Telescoping or Non-Standard Summation Involving an AP View

For an arithmetic sequence $\{a_n\}$ with nonzero common difference, $$|a_6| = a_8, \quad \sum_{k=1}^{5} \frac{1}{a_k a_{k+1}} = \frac{5}{96}$$ Find the value of $\sum_{k=1}^{15} a_k$. [4 points]
(1) 60
(2) 65
(3) 70
(4) 75
(5) 80

csat-suneung 2024 Q18 3 marks Compute Partial Sum of an Arithmetic Sequence View

For two sequences $\{a_n\}$ and $\{b_n\}$, $$\sum_{k=1}^{10} a_k = \sum_{k=1}^{10} (2b_k - 1), \quad \sum_{k=1}^{10} (3a_k + b_k) = 33$$ Find the value of $\sum_{k=1}^{10} b_k$. [3 points]

csat-suneung 2025 Q12 4 marks Summation of Derived Sequence from AP View

A sequence $\left\{ a_{n} \right\}$ with $a_{1} = 2$ and an arithmetic sequence $\left\{ b_{n} \right\}$ with $b_{1} = 2$ satisfy $$\sum_{k=1}^{n} \frac{a_{k}}{b_{k+1}} = \frac{1}{2}n^{2}$$ for all natural numbers $n$. What is the value of $\sum_{k=1}^{5} a_{k}$? [4 points]
(1) 120
(2) 125
(3) 130
(4) 135
(5) 140

csat-suneung 2025 Q18 3 marks Compute Partial Sum of an Arithmetic Sequence View

A sequence $\left\{ a_{n} \right\}$ satisfies $$a_{n} + a_{n+4} = 12$$ for all natural numbers $n$. What is the value of $\sum_{n=1}^{16} a_{n}$? [3 points]

csat-suneung 2026 Q2 3 marks Compute Partial Sum of an Arithmetic Sequence View

For the sequence $\left\{ a _ { n } \right\}$, when $\sum _ { k = 1 } ^ { 4 } \left( 2 a _ { k } - k \right) = 0$, what is the value of $\sum _ { k = 1 } ^ { 4 } a _ { k }$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

gaokao 2015 Q2 Find Specific Term from Given Conditions View

2. In the arithmetic sequence $\left\{ a _ { n } \right\}$, if $a _ { 2 } = 4 , a _ { 4 } = 2$, then $a _ { 6 } =$
A. $-1$
B. $0$
C. $1$
D. $6$

gaokao 2015 Q3 Arithmetic-Geometric Hybrid Problem View

3. Given that $\left\{ a _ { n } \right\}$ is an arithmetic sequence with non-zero common difference $d$, and the sum of the first $n$ terms is $S _ { n }$. If $a _ { 3 } , a _ { 4 } , a _ { 8 }$ form a geometric sequence, then
A. $a _ { 1 } d > 0 , d S _ { n } > 0$
B. $a _ { 1 } d < 0 , d S _ { n } < 0$
C. $a _ { 1 } d > 0 , d S _ { n } < 0$
D. $a _ { 1 } d < 0 , d S _ { n } > 0$ [Figure]

gaokao 2015 Q5 Flowchart or Algorithm Tracing Involving Sequences View

5. Executing the flowchart shown in Figure 2, if the input is $n = 3$, then the output $S =$
A. $\frac { 6 } { 7 }$
B. $\frac { 3 } { 7 }$
C. $\frac { 8 } { 9 }$
D. $\frac { 4 } { 9 }$
[Figure]
Figure 2

gaokao 2015 Q5 Compute Partial Sum of an Arithmetic Sequence View

5. Let $S _ { n }$ be the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$. If $a _ { 1 } + a _ { 3 } + a _ { 5 } = 3$, then $S _ { 5 } =$ [Figure] [Figure]
A. $5$
B. $7$
C. $9$
D. $11$

gaokao 2015 Q6 Properties of AP Terms under Transformation View

6. Let $\left\{ a _ { n } \right\}$ be an arithmetic sequence. The correct conclusion is
A. If $a _ { 1 } + a _ { 2 } > 0$, then $a _ { 2 } + a _ { 3 } > 0$
B. If $a _ { 1 } + a _ { 3 } < 0$, then $a _ { 1 } + a _ { 2 } < 0$
C. If $0 < a _ { 1 } < a _ { 2 }$, then $a _ { 2 } > \sqrt { a _ { 1 } a _ { 3 } }$
D. If $a _ { 1 } < 0$, then $\left( a _ { 2 } - a _ { 1 } \right) \left( a _ { 2 } - a _ { 3 } \right) > 0$

gaokao 2015 Q7 Flowchart or Algorithm Tracing Involving Sequences View

7. Executing the flowchart shown in question (7), if the input value of K is 8, then the condition that can be filled in the decision box is
A. $\mathrm { s } \leq \frac { 3 } { 4 }$
B. $\mathrm { s } \leq \frac { 5 } { 6 }$
C. $\mathrm { s } \leq \frac { 11 } { 12 }$
D. $\mathrm { s } \leq \frac { 15 } { 24 }$ [Figure]

gaokao 2015 Q8 5 marks Flowchart or Algorithm Tracing Involving Sequences View

Executing the flowchart shown in figure (8), the output value of $s$ is
(A) $\frac { 3 } { 4 }$
(B) $\frac { 5 } { 6 }$
(C) $\frac { 11 } { 12 }$
(D) $\frac { 25 } { 24 }$

gaokao 2015 Q10 Arithmetic-Geometric Hybrid Problem View

10. Given that $\left\{ a _ { n } \right\}$ is an arithmetic sequence with non-zero common difference $d$. If $a _ { 2 } , a _ { 3 } , a _ { 7 }$ form a geometric sequence, and $2 a _ { 1 } + a _ { 2 } = 1$ , then $a _ { 1 } =$ $\_\_\_\_$ , $d =$ $\_\_\_\_$.

gaokao 2015 Q11 Telescoping or Non-Standard Summation Involving an AP View

11. The sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } = 1$ and $a _ { n + 1 } - a _ { n } = n + 1 \quad \left( n \in N ^ { * } \right)$, then the sum of the first 10 terms of the sequence $\left\{ \frac { 1 } { a _ { n } } \right\}$ is $\_\_\_\_$.

gaokao 2015 Q13 Find Specific Term from Given Conditions View

13. A group of 1010 numbers with median 1010 form an arithmetic sequence with last term 2015. The first term of this sequence is $\_\_\_\_$

gaokao 2015 Q16 13 marks Multi-Part Structured Problem on AP View

Given that the arithmetic sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } + a _ { 2 } = 10$ and $a _ { 4 } - a _ { 3 } = 2$.\n(I) Find the general term formula of $\left\{ a _ { n } \right\}$;\n(II) Let the geometric sequence $\left\{ b _ { n } \right\}$ satisfy $b _ { 2 } = a _ { 3 }$ and $b _ { 3 } = a _ { 7 }$. Question: Which term of the sequence $\left\{ a _ { n } \right\}$ is equal to $b _ { 6 }$?

gaokao 2015 Q16 13 marks Multi-Part Structured Problem on AP View

An arithmetic sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 3 } = 2$ and the sum of the first 3 terms $S _ { 3 } = \frac { 9 } { 2 }$ .
(I) Find the general term formula for $\left\{ a _ { n } \right\}$;
(II) A geometric sequence $\left\{ b _ { n } \right\}$ satisfies $b _ { 1 } = a _ { 1 } , b _ { 4 } = a _ { 15 }$. Find the sum of the first $n$ terms $T _ { n }$ of $\left\{ b _ { n } \right\}$.

gaokao 2015 Q17 15 marks Multi-Part Structured Problem on AP View

17. (15 points) Given sequences $\left\{ a _ { n } \right\}$ and $\left\{ b _ { n } \right\}$ satisfying $a _ { 1 } = 2 , b _ { 1 } = 1 , a _ { n + 1 } = 2 a _ { n } \left( \mathrm { n } \in \mathrm { N } ^ { * } \right)$ , $b _ { 1 } + \frac { 1 } { 2 } b _ { 2 } + \frac { 1 } { 3 } b _ { 3 } + \cdots + \frac { 1 } { n } b _ { n } = b _ { n + 1 } - 1 \left( \mathrm { n } \in \mathrm { N } ^ { * } \right)$ .
(1) Find $a _ { n }$ and $b _ { n }$ ;
(2) Let $T _ { n }$ denote the sum of the first n terms of the sequence $\left\{ a _ { n } b _ { n } \right\}$ . Find $T _ { n }$ .

gaokao 2015 Q18 Arithmetic-Geometric Hybrid Problem View

18. Given that $\{ a _ { n } \}$ is a geometric sequence with all positive terms, $\{ b _ { n } \}$ is an arithmetic sequence, and $a _ { 1 } = b _ { 1 } = 1$, $b _ { 2 } + b _ { 3 } = 2 a _ { 3 }$, $a _ { 5 } - 3 b _ { 2 } = 7$.
(1) Find the general term formulas for $\{ a _ { n } \}$ and $\{ b _ { n } \}$;
(2) Let $c _ { n } = a _ { n } b _ { n } , n \in \mathbb{N} ^ { * }$. Find the sum of the first $n$ terms of the sequence $\{ c _ { n } \}$.

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LFM Pure and Mechanics

Indices and Surds 107

Exponential Functions 100

Laws of Logarithms 200

Exponential Equations & Modelling 58

Arithmetic Sequences and Series 271

Geometric Sequences and Series 203

Differentiation from First Principles 37

Tangents, normals and gradients 93

Stationary points and optimisation 384

Applied differentiation 164

Indefinite & Definite Integrals 533

Areas Between Curves 83

Areas by integration 109

Volumes of Revolution 61

Numerical integration 32

Vectors Introduction & 2D 204

Vectors 3D & Lines 233

Travel graphs 11

SUVAT in 2D & Gravity 6

Constant acceleration (SUVAT) 32

Variable acceleration (1D) 40

Variable acceleration (vectors) 26

Forces, equilibrium and resultants 9

Newton's laws and connected particles 18

Pulley systems 5

Motion on a slope 4

Friction 10

Momentum and Collisions 15

Moments 13

Parametric curves and Cartesian conversion 11

Parametric differentiation 15

Parametric integration 4

Projectiles 40