IGCSE Mathematics Workbook 100: Number Patterns and Sequences | IGCSE数学练习册100：数字规律与数列

📚 IGCSE Mathematics Workbook 100: Number Patterns and Sequences | IGCSE数学练习册100：数字规律与数列

Welcome to Workbook 100 of the TutorHao IGCSE Mathematics revision series. This workbook focuses on one of the most fundamental topics in the IGCSE syllabus: number patterns and sequences. Mastering this area not only strengthens your algebraic skills but also builds logical thinking that is essential for higher-level mathematics.

欢迎来到 TutorHao IGCSE 数学复习系列的第 100 本练习册。本练习册聚焦于 IGCSE 考纲中最基础的主题之一：数字规律与数列。掌握这一领域不仅能增强你的代数能力，还能培养对更高阶数学至关重要的逻辑思维。

1. What Is a Sequence? | 什么是数列？

A sequence is an ordered list of numbers that follow a specific rule. Each number in the sequence is called a term, and the position of a term is indicated by a positive integer n. For example, in the sequence 2, 4, 6, 8, 10, the first term is 2, the second term is 4, and so on.

数列是按照特定规则排列的一组有序数字。数列中的每个数字称为一项，项的位置用正整数 n 表示。例如，在数列 2, 4, 6, 8, 10 中，第一项是 2，第二项是 4，以此类推。

There are two main types of rules that define a sequence: a term-to-term rule, which tells you how to move from one term to the next, and a position-to-term rule, which gives a formula to calculate any term directly from its position.

定义数列的规则主要有两种：逐项递推规则，它告诉你如何从一项得到下一项；以及项位对应规则，它给出一个公式，可以直接由项的位置计算出该项的值。

2. Arithmetic Sequences | 等差数列

An arithmetic sequence is a sequence where the difference between consecutive terms is constant. This constant is called the common difference, usually denoted by d. For instance, 3, 7, 11, 15, 19 is arithmetic because each term increases by 4.

等差数列是指相邻项之间的差为常数的数列。这个常数称为公差，通常用 d 表示。例如，3, 7, 11, 15, 19 是等差数列，因为每一项增加 4。

n-th term = a + (n − 1) × d

Here, a is the first term, n is the term number, and d is the common difference. Using this formula, you can find any term without listing the whole sequence.

其中 a 是首项，n 是项数，d 是公差。使用这个公式，你可以不用列出整个数列就能求出任意一项。

3. Finding the nth Term of an Arithmetic Sequence | 求等差数列的第 n 项

Consider the sequence 5, 8, 11, 14, 17. The first term a = 5, and the common difference d = 3. To find the 10th term, substitute n = 10 into the formula:

考虑数列 5, 8, 11, 14, 17。首项 a = 5，公差 d = 3。要求第 10 项，将 n = 10 代入公式：

10th term = 5 + (10 − 1) × 3 = 5 + 27 = 32

Therefore, the 10th term is 32. This direct method is fast and reliable, especially when the position n is large.

因此，第 10 项为 32。这种直接方法快速且可靠，尤其当 n 很大时非常高效。

Always identify a and d carefully before substituting. A common mistake is to forget that (n − 1), not n, must be multiplied by d.

代入前务必仔细识别 a 和 d。一个常见错误是忘记与 d 相乘的应该是 (n − 1) 而非 n。

4. Geometric Sequences | 等比数列

A geometric sequence is a sequence where each term is obtained by multiplying the previous term by a constant ratio, called the common ratio r. For example, 2, 6, 18, 54, 162 is geometric because each term is multiplied by 3.

等比数列是指每一项通过将前一项乘以一个固定比例得到的数列，这个比例称为公比 r。例如，2, 6, 18, 54, 162 是等比数列，因为每项都乘以 3。

n-th term = a × rⁿ⁻¹

Here, a is the first term and n is the term number. This formula is useful for both finding specific terms and solving problems involving growth or decay.

其中 a 是首项，n 是项数。这个公式既适用于求具体项，也适用于解决涉及增长或衰减的问题。

Remember that r can be a fraction or a negative number. For instance, a ratio of ½ produces a decreasing sequence, while a negative ratio produces terms that alternate in sign.

记住 r 可以是分数或负数。例如，公比为 ½ 时数列递减，而公比为负数时各项符号交替变化。

5. Square Numbers, Cube Numbers, and Triangular Numbers | 平方数、立方数与三角数

Some important sequences appear frequently in IGCSE exams. The sequence of square numbers is 1, 4, 9, 16, 25, … and its n-th term is n². The sequence of cube numbers is 1, 8, 27, 64, 125, … with n-th term n³.

一些重要的数列在 IGCSE 考试中经常出现。平方数列为 1, 4, 9, 16, 25, …，其第 n 项为 n²。立方数列为 1, 8, 27, 64, 125, …，其第 n 项为 n³。

Triangular numbers are built by adding consecutive whole numbers: 1, 3, 6, 10, 15, … The n-th triangular number is given by:

三角数通过累加连续整数得到：1, 3, 6, 10, 15, …。第 n 个三角数的公式为：

Tₙ = n × (n + 1) ÷ 2

For example, the 6th triangular number is 6 × 7 ÷ 2 = 21. These special sequences appear in many exam questions, so memorising their formulas is essential.

例如，第 6 个三角数为 6 × 7 ÷ 2 = 21。这些特殊数列在许多考试题目中出现，因此牢记其公式至关重要。

6. Fibonacci-Style Sequences | 斐波那契型数列

In a Fibonacci-style sequence, each term is the sum of the two previous terms. The classic Fibonacci sequence starts with 1, 1, 2, 3, 5, 8, 13, … where the third term is 1 + 1 = 2, the fourth is 1 + 2 = 3, and so on.

在斐波那契型数列中，每一项等于前两项之和。经典斐波那契数列始于 1, 1, 2, 3, 5, 8, 13, …，其中第三项为 1 + 1 = 2，第四项为 1 + 2 = 3，以此类推。

In IGCSE questions, you may be given a different starting point. For example, if the first two terms are 2 and 3, the sequence becomes 2, 3, 5, 8, 13, 21, … The rule remains the same: termₙ = termₙ₋₁ + termₙ₋₂ for n ≥ 3.

在 IGCSE 题目中，你可能会得到不同的起始项。例如，若前两项为 2 和 3，数列变为 2, 3, 5, 8, 13, 21, …。规则不变：第 n 项 = 第(n−1)项 + 第(n−2)项，其中 n ≥ 3。

When solving Fibonacci problems, always write down the terms step by step. Missing one term can cause your entire sequence to be wrong.

在解答斐波那契问题时，务必一步一步写下各项。漏掉任何一项都可能导致整个数列错误。

7. Position-to-Term Rules vs Term-to-Term Rules | 项位对应规则与逐项递推规则

A position-to-term rule gives a formula for the n-th term directly. For example, the sequence 4, 7, 10, 13 has the rule 3n + 1. A term-to-term rule, on the other hand, describes how to go from one term to the next, such as “add 3 each time.”

项位对应规则直接给出第 n 项的公式。例如，数列 4, 7, 10, 13 的规则为 3n + 1。而逐项递推规则描述如何从前一项得到后一项，如“每次加 3”。

In exams, you might be asked to convert one form into the other. To find a position-to-term rule from a term-to-term rule, you still need the first term and the common difference.

在考试中，你可能会被要求将一种形式转换为另一种。要从逐项递推规则求出项位对应规则，你仍然需要首项和公差。

Type	Example rule	Typical question
Position-to-term	n-th term = 3n + 1	Find the 20th term
Term-to-term	Add 3 to the previous term	Write the first 5 terms

8. Practical Problems with Sequences | 数列的实际应用问题

Sequences are not just abstract exercises; they appear in real-life situations such as saving money, housing patterns, or plant growth. For example, suppose you save $10 in the first week and increase your savings by $5 each week. The amounts form the arithmetic sequence 10, 15, 20, 25, …

数列并非只是抽象练习，它们出现在实际生活情境中，例如存钱、住房布局或植物生长。例如，假设你第一周存 10 美元，之后每周多存 5 美元。存款金额构成等差数列 10, 15, 20, 25, …。

To find how much you save in the 12th week, use the formula:

要求第 12 周存多少钱，使用公式：

12th term = 10 + (12 − 1) × 5 = 10 + 55 = 65

So you would save $65 in the 12th week. Always define your variables clearly in word problems and check whether the sequence is arithmetic or geometric.

因此，你第 12 周将存 65 美元。在文字题中务必清晰定义变量，并检查数列是等差数列还是等比数列。

9. Common Mistakes to Avoid | 常见错误警示

Many students lose marks on sequence questions because of small errors. The most frequent mistakes are using n instead of (n − 1) in the arithmetic formula, confusing the common difference with the common ratio, and forgetting to simplify algebraic expressions.

许多学生在数列题目中因小错误而失分。最常见的错误包括：在等差数列公式中使用 n 而非 (n − 1)，混淆公差与公比，以及忘记化简代数表达式。

Always read the question carefully to determine whether it asks for the first term or the n-th term.
为避免此类错误，务必仔细阅读题目，判断要求的是首项还是第 n 项。
Check if the sequence increases by addition (arithmetic) or multiplication (geometric).
检查数列是通过加法（等差）递增还是通过乘法（等比）递增。
Verify your formula with at least one known term before answering.
作答前，至少用一个已知项验证你的公式。

10. Exam-Style Questions | 考试题型演练

Here is a typical IGCSE question: A sequence has its n-th term given by 2n² − 3. Find the first four terms and the 10th term. Substituting n = 1, 2, 3, 4 gives −1, 5, 15, 29, and for n = 10 you get 2(100) − 3 = 197.

以下是一道典型 IGCSE 题目：某数列的第 n 项由 2n² − 3 给出。求前四项及第 10 项。代入 n = 1, 2, 3, 4 得到 −1, 5, 15, 29；代入 n = 10 得 2(100) − 3 = 197。

Another common type asks you to identify the rule from a given sequence. For the sequence 3, 9, 27, 81, the pattern is “multiply by 3”, so the n-th term is 3 × 3ⁿ⁻¹, which simplifies to 3ⁿ.

另一种常见题型是要求你根据给定数列识别规则。对于数列 3, 9, 27, 81，规律是“乘以 3”，因此第 n 项为 3 × 3ⁿ⁻¹，化简为 3ⁿ。

Practice rewriting fractions and negative terms carefully. Accuracy in substitution is the key to full marks.

练习时仔细处理分数和负项。代入的准确性是获取满分的关键。

11. Revision Tips for Sequence Topics | 数列专题复习建议

To master sequences, start by memorising the basic formulas for arithmetic and geometric sequences. Then, practise identifying the type of sequence quickly by looking at the differences or ratios between consecutive terms.

要掌握数列，首先牢记等差数列与等比数列的基本公式。然后通过观察相邻项的差或比，练习快速判断数列类型。

Create a formula sheet with the n-th term formulas for squares, cubes, and triangular numbers.
制作一张公式表，列出平方数、立方数和三角数的第 n 项公式。
Solve at least five exam-style problems daily during revision weeks.
复习周期间每天至少完成五道考试题型练习。
Explain the concept to a friend; teaching others helps solidify understanding.
向朋友讲解概念；教他人有助于巩固理解。

12. Final Checklist for Workbook 100 | 练习册 100 最终检查清单

Before moving on to the next workbook, make sure you can answer ‘yes’ to each of the following questions. This checklist summarises the core skills covered in Workbook 100.

在进入下一本练习册之前，请确保你能对以下每个问题回答“是”。这份检查清单总结了练习册 100 的核心技能。

Can you define arithmetic and geometric sequences clearly?
你能清晰定义等差数列和等比数列吗？
Can you find the n-th term of a given arithmetic sequence using the formula a + (n − 1) × d?
你能使用公式 a + (n − 1) × d 求给定等差数列的第 n 项吗？
Can you determine the common ratio of a geometric sequence?
你能确定等比数列的公比吗？
Can you work with square, cube, and triangular number sequences?
你能处理平方数、立方数和三角数数列吗？
Can you solve word problems involving sequences step by step?
你能逐步解决与数列相关的应用问题吗？
Can you identify and avoid the common mistakes listed in Section 9?
你能识别并避免第 9 节列出的常见错误吗？

If you answered ‘yes’ to all questions, congratulations! You have completed Workbook 100. Remember that consistent practice is the most reliable way to build confidence for your IGCSE mathematics examination.

如果你对所有问题都回答了“是”，恭喜你！你已完成练习册 100。请记住，持续练习是建立 IGCSE 数学考试信心的最可靠途径。

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IGCSE Mathematics Workbook 100: Number Patterns and Sequences | IGCSE数学练习册100：数字规律与数列