📚 Number 8: Sequences | 序列
A sequence is an ordered list of numbers following a specific rule. In IGCSE Edexcel Mathematics, sequences form a core part of the number topic. You must be able to recognise linear, quadratic, geometric and Fibonacci-type sequences, find the nth term, use recurrence relations, and apply sequences in real-world contexts. This article covers every key skill and common exam pitfall.
序列是按特定规律排列的一列数。在 Edexcel IGCSE 数学中，序列是数论部分的核心内容。你需要识别线性序列、二次序列、等比序列和斐波那契型序列，求出第 n 项，使用递推关系，并在实际情境中应用序列。本文涵盖所有关键技能和常见考试陷阱。
1. What Are Sequences? | 什么是序列？
or infinite (continuing forever). The position of a term is given by n, where n = 1, 2, 3, …
序列是一组按确定顺序排列的数。序列中的每一个数称为一项。序列可以是有限的（项数有限）或无限的（永远延续下去）。项的位置用 n 表示，n = 1, 2, 3, …
The rule that generates the sequence may be given as a formula for the nth term (position-to-term rule) or as a relationship connecting consecutive terms (term-to-term rule or recurrence relation).
生成序列的规则可以表示为第 n 项的公式（定位法则），也可以表示为相邻项之间的关系（项项法则或递推关系）。
- Position-to-term: uₙ = 3n + 2 → terms 5, 8, 11, 14, …
- 定位法则：uₙ = 3n + 2 → 各项为 5, 8, 11, 14, …
- Term-to-term: start with 5, then add 3 each time
- 项项法则：从 5 开始，每次都加 3
2. Arithmetic (Linear) Sequences | 等差（线性）序列
An arithmetic sequence, also called a linear sequence, has a constant difference between consecutive terms. This difference is known as the common difference, d. The first term is denoted by a.
等差数列又称线性序列，相邻两项之间的差是常数。这个差称为公差 d。首项用 a 表示。
For example, in the sequence 7, 10, 13, 16, … the first term a = 7 and the common difference d = +3.
例如，在序列 7, 10, 13, 16, … 中，首项 a = 7，公差 d = +3。
The nth term formula is:
第 n 项公式为：
uₙ = a + (n – 1)d
You can use this to find any term directly without listing all previous terms.
你可以用它直接求出任意一项，而无需列出前面所有项。
3. Finding the nth Term of a Linear Sequence | 求线性序列的第 n 项
becomes the coefficient of n. Then adjust by comparing with n=1 to find the constant term.
求线性序列的第 n 项时，首先要找到公差 d，它成为 n 的系数。然后代入 n=1 进行比较，求出常数项。
Steps:
步骤：
- Find d from the first two terms. Example: terms 4, 9, 14, 19 → d = 5.
- 由前两项求出 d。例如：项 4, 9, 14, 19 → d = 5。
- Write uₙ = 5n + c. Substitute n=1: 5(1)+c = 4 → c = –1.
- 写出 uₙ = 5n + c。代入 n=1：5(1)+c = 4 → c = –1。
- So uₙ = 5n – 1. Check for n=3: 5(3)–1 = 14, correct.
- 因此 uₙ = 5n – 1。检验 n=3：5(3)–1 = 14，正确。
Always test your nth term with n=2 or n=3 to be safe.
为确保正确，务必用 n=2 或 n=3 检验你的第 n 项公式。
4. Quadratic Sequences | 二次序列
olves n². The general form is uₙ = an² + bn + c, where a, b and c are numbers to be found.
二次序列具有恒定的二级差分。第 n 项包含 n²，其一般形式为 uₙ = an² + bn + c，其中 a、b 和 c 是待求的常数。
| Sequence: 3, 9, 19, 33, 51 |
| First differences: +6, +10, +14, +18 |
| Second differences: +4, +4, +4 (constant) |
Since the second difference is 4, then 2a = 4 → a = 2.
由于二级差分为 4，所以 2a = 4 → a = 2。
5. Finding the nth Term of a Quadratic Sequence | 求二次序列的第 n 项
Once you know a, subtract the sequence an² from the original sequence to obtain a linear remainder. Then find the nth term of that linear part.
一旦求得 a，从原序列中减去序列 an²，得到线性余项，再求该线性部分的第 n 项。
Example: Find the nth term of 3, 9, 19, 33, 51.
示例：求序列 3, 9, 19, 33, 51 的第 n 项。
Step 1: Second difference = 4 → 2a=4 → a=2. So the 2n² terms: 2, 8, 18, 32, 50.
步骤1：二级差 = 4 → 2a=4 → a=2。因此 2n² 项为：2, 8, 18, 32, 50。
Step 2: Subtract: 3–2=1, 9–8=1, 19–18=1, 33–32=1, 51–50=1 → linear remainder: 1, 1, 1, 1, 1 → constant sequence, so b=0, c=1.
步骤2：相减：3–2=1，9–8=1，19–18=1，33–32=1，51–50=1 → 线性余项：1, 1, 1, 1, 1 → 常数序列，所以 b=0, c=1。
Thus uₙ = 2n² + 1.
因此 uₙ = 2n² + 1。
If the linear remainder is not constant, find its nth term as bn + c by the standard linear method.
如果线性余项不是常数，则按标准线性方法求其第 n 项 bn + c。
6. Geometric Sequences | 等比序列
A geometric sequence multiplies by a constant factor, called the common ratio, r, to get from one term to the next. The first term is a.
等比序列通过乘以一个常数因子——公比 r，从前一项得到后一项。首项为 a。
Example: 5, 15, 45, 135, … has a = 5, r = 3.
示例：5, 15, 45, 135, … 中 a = 5, r = 3。
The nth term is:
第 n 项公式为：
uₙ = arⁿ⁻¹
Using the example, u₄ = 5 × 3⁴⁻¹ = 5 × 27 = 135. Geometric sequences grow rapidly when |r| > 1.
用此示例，u₄ = 5 × 3⁴⁻¹ = 5 × 27 = 135。当 |r| > 1 时，等比序列增长迅速。
Be careful: r can be a fraction, giving a decreasing sequence, e.g. 80, 40, 20, 10, … has r = ½.
注意：r 可为分数，此时序列递减，例如 80, 40, 20, 10, … 的公比 r = ½。
7. Recurrence Relations | 递推关系
A recurrence relation defines the next term using one or more preceding terms. This is a term-to-term rule, often used when the nth term is not given explicitly.
递推关系利用前一项或前几项来定义下一项，是一种项项法则，常用于没有直接给出第 n 项公式的情况。
Form: uₙ₊₁ = uₙ + d (for arithmetic) or uₙ₊₁ = 2uₙ – 3, etc., with given u₁.
形式：uₙ₊₁ = uₙ + d（等差）或 uₙ₊₁ = 2uₙ – 3 等，并给出 u₁。
Example: u₁ = 4, uₙ₊₁ = 3uₙ – 2. Then u₂ = 3(4)–2 = 10; u₃ = 3(10)–2 = 28; u₄ = 3(28)–2 = 82.
示例：u₁ = 4，uₙ₊₁ = 3uₙ – 2。则 u₂ = 3(4)–2 = 10；u₃ = 3(10)–2 = 28；u₄ = 3(28)–2 = 82。
Recurrence relations appear frequently in IGCSE; you must be able to generate terms and sometimes link back to explicit nth term formulas.
递推关系在 IGCSE 中经常出现，你必须能够生成各项，有时还需将其与显式的第 n 项公式关联起来。
8. Fibonacci-type Sequences | 斐波那契型序列
A Fibonacci-type sequence is defined by adding the two previous terms to get the next term. The classic Fibonacci sequence starts 1, 1, 2, 3, 5, 8, 13, …
斐波那契型序列通过将前两项相加得到下一项。经典斐波那契序列从 1, 1, 2, 3, 5, 8, 13, … 开始。
uₙ with u₁ and u₂ given. Edexcel might ask you to generate terms or recognise the pattern.
递推规则为 uₙ₊₂ = uₙ₊₁ + uₙ，并给出 u₁ 和 u₂。Edexcel 考试可能会要求你生成各项或识别该模式。
You could be given different starting values, e.g. 3 and 4 → 3, 4, 7, 11, 18, … Always read the question carefully.
你可能会遇到不同的初始值，例如 3 和 4 → 3, 4, 7, 11, 18, … 务必仔细读题。
9. Special Sequences and Real-life Contexts | 特殊序列与实际情境
Triangular numbers: 1, 3, 6, 10, 15, … Their nth term is Tₙ = n(n+1)/2. Square numbers: 1, 4, 9, 16, 25, … (uₙ = n²). These often appear in pattern-based problems.
三角形数：1, 3, 6, 10, 15, … 其第 n 项为 Tₙ = n(n+1)/2。平方数：1, 4, 9, 16, 25, … (uₙ = n²)。它们常出现在图形规律题中。
Sequences can model real situations such as savings with simple or compound interest, tile patterns, seating arrangements, or bouncing ball heights (geometric).
序列可以模拟实际情境，如单利或复利储蓄、瓷砖图形、座位安排，或弹跳球高度（等比序列）。
n row 3, and so on. This is an arithmetic sequence with a=20, d=4; the nth row has 20 + (n–1)×4 seats.
示例：某影院第 1 排有 20 个座位，第 2 排有 24 个，第 3 排有 28 个，以此类推。这是一个等差序列，a=20, d=4；第 n 排有 20 + (n–1)×4 个座位。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Mistake 1: Confusing linear and quadratic sequences. Always check whether the first differences are constant or the second differences.
错误1：混淆线性序列和二次序列。务必检查一级差分是否恒定，或二级差分是否恒定。
ula with the 3rd term or beyond. Testing only n=1 and n=2 can still hide a wrong constant.
错误2：忘记用第 3 项或更后面项检验第 n 项公式。只用 n=1 和 n=2 检验可能仍然隐藏着错误的常数项。
Mistake 3: In geometric sequences, using uₙ = arⁿ instead of arⁿ⁻¹. The exponent must be n–1 because the first term has no multiplication by r yet.
错误3：在等比序列中，误用 uₙ = arⁿ 而非 arⁿ⁻¹。指数必须是 n–1，因为首项尚未乘以 r。
Mistake 4: Misreading the starting index. Some questions use zero-based indexing (n=0,1,2,…). Read the given pattern carefully.
错误4：看错起始下标。有些问题采用 n=0 起始。仔细阅读所给规律。
Tip: Show your second difference calculation for quadratic sequences to earn method marks even if you make a slip later.
技巧：展示二次序列的二级差分计算过程，这样即使后续计算出错也能获得方法分。
11. Summary | 总结
Mastering sequences in IGCSE Edexcel Mathematics requires fluency with linear, quadratic, geometric and Fibonacci-type rules. Remember the key formulas: uₙ = a + (n–1)d for arithmetic sequences, uₙ = arⁿ⁻¹ for geometric sequences, and the second-difference method for quadratics. Practise reading both position-to-term and term-to-term descriptions, and always check your answers. Sequences thinking will also support your algebra and problem-solving skills throughout the syllabus.
精通 Edexcel IGCSE 数学中的序列，需要熟练掌握线性、二次、等比和斐波那契型规律。记住核心公式：等差序列的 uₙ = a + (n–1)d，等比序列的 uₙ = arⁿ⁻¹，以及二次序列的二级差分法。练习阅读定位法则和项项法则，并务必检验答案。序列思维还将助力你在整个大纲中的代数与问题求解能力。
Published by TutorHao | Number Revision Series | aleveler.com
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