📚 Number Patterns and Sequences | 数字规律与数列
Patterns are the foundation of algebra. In IGCSE Mathematics you must be able to continue a sequence, describe the rule that generates it, and find any term from its position.
规律是代数的基础。在 IGCSE 数学中，你必须能够续写数列、描述生成数列的规则，并根据项的位置求出任意一项。
1. Understanding Number Patterns | 理解数字规律
A number pattern is an ordered list of numbers that follows a clear rule. The individual numbers in the list are called terms.
数字规律是一组按照明确规则排列的有序数字。列表中的每一个数字称为项。
Example: 4, 8, 12, 16, 20 ... has the rule "add 4 each time".
例：4、8、12、16、20…… 的规则是“每次加 4”。
| Position n | 位置 n | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Term | 项 | 4 | 8 | 12 | 16 | 20 |
2. Term-to-Term Rules | 逐项变化规则
A term-to-term rule tells you how to move from one term to the next. It is usually found by subtracting two consecutive terms.
逐项变化规则告诉你如何从一项得到下一项。通常可以通过相邻两项相减来找到它。
For example, in 7, 11, 15, 19, 23, the difference between consecutive terms is constant:
例如，在 7、11、15、19、23 中，相邻项之间的差是固定的：
11 − 7 = 4, 15 − 11 = 4, 19 − 15 = 4
So the term-to-term rule is "add 4".
因此逐项变化规则是“加 4”。
3. Input and Output Machines | 输入与输出机器
A number pattern can be produced by an input-output machine. The input is the position number n, and the output is the term itself.
数字规律可以通过输入输出机器产生。输入是位置数 n，输出是对应的项。
For the sequence 3, 5, 7, 9, 11, the machine rule could be "multiply n by 2, then add 1".
对于数列 3、5、7、9、11，机器规则可以是“将 n 乘以 2，再加 1”。
| Input n | 输入 n | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Output term | 输出项 | 3 | 5 | 7 | 9 | 11 |
This position-to-term rule is more powerful because it allows you to jump directly to any term without listing the whole sequence.
位置与项的关系规则更强大，因为它允许你直接跳到任意一项，而无需列出整个数列。
4. Arithmetic Sequences | 等差数列
An arithmetic sequence is a sequence where the difference between any two consecutive terms is always the same. This fixed difference is called the common difference, d.
等差数列是指任意两个相邻项之差始终相同的数列。这个固定的差称为公差 d。
For example, 9, 13, 17, 21, 25 is an arithmetic sequence with d = 4.
例如，9、13、17、21、25 是一个公差 d = 4 的等差数列。
aₙ = a₁ + (n − 1)d
Here aₙ is the nth term, a₁ is the first term, n is the position, and d is the common difference.
这里 aₙ 是第 n 项，a₁ 是首项，n 是位置，d 是公差。
5. Finding the nth Term | 求第 n 项
To find the nth term of an arithmetic sequence, first identify the first term a₁ and the common difference d.
要求等差数列的第 n 项，首先要确定首项 a₁ 和公差 d。
Consider the sequence 5, 8, 11, 14, 17 ...
考虑数列 5、8、11、14、17……
Here a₁ = 5 and d = 3, so:
这里 a₁ = 5，d = 3，因此：
aₙ = 5 + (n − 1) × 3
Simplify this bracket:
化简括号：
aₙ = 5 + 3n − 3 = 3n + 2
So the nth term is 3n + 2. The 50th term is 3 × 50 + 2 = 152.
因此第 n 项是 3n + 2。第 50 项为 3 × 50 + 2 = 152。
6. Decreasing Sequences | 递减数列
Not all arithmetic sequences increase. If the common difference is negative, the sequence decreases.
并非所有等差数列都是递增的。如果公差为负数，数列就会递减。
For example, 23, 19, 15, 11, 7 has a common difference d = −4.
例如，23、19、15、11、7 的公差 d = −4。
The nth term is still found using the same formula:
仍然使用同一公式求第 n 项：
aₙ = 23 + (n − 1) × (−4)
Simplify to get:
化简得到：
aₙ = 23 − 4n + 4 = 27 − 4n
This shows the coefficient of n is negative for a decreasing arithmetic sequence.
这表明递减等差数列中 n 的系数为负数。
7. Using a Difference Table | 用差分表找规律
When you are given only the first few terms, a difference table helps you spot the rule quickly.
当你只给出前几项时，差分表可以帮助你快速发现规律。
For the sequence 6, 10, 14, 18, 22, write the differences below the terms:
对于数列 6、10、14、18、22，把差值写在项的下面：
| Term | 项 | 6 | 10 | 14 | 18 | 22 |
|---|---|---|---|---|---|
| Difference | 差 | 4 | 4 | 4 | 4 |
Since the first difference is constant at 4, the sequence is arithmetic and has nth term 4n + 2.
由于一阶差分恒为 4，该数列是等差数列，其第 n 项为 4n + 2。
8. Patterns in Diagrams and Shapes | 图形与图表中的规律
IGCSE problems often link number sequences to patterns made from dots, sticks, or tiles.
IGCSE 题目经常将数列与由点、棍或瓷砖组成的图形规律联系起来。
For example, a pattern uses matchsticks to form squares. Each new square adds 3 matchsticks, so the sequence 4, 7, 10, 13 ... appears.
例如，用火柴棒拼正方形的规律中，每增加一个正方形就增加 3 根火柴棒，因此出现数列 4、7、10、13……
The number of matchsticks for n squares is given by the nth term 3n + 1.
n 个正方形所需的火柴棒数由第 n 项 3n + 1 给出。
This connects a visual pattern to an algebraic rule, which is a key IGCSE skill.
这将图形规律与代数规则联系起来，是 IGCSE 数学的关键技能。
9. Problem-Solving Strategy: Look for a Pattern | 问题解决策略：寻找规律
When a problem looks complicated, writing the first few values and looking for a pattern can reveal a simple rule.
当问题看起来复杂时，写出前几个数值并寻找规律可以揭示简单的规则。
Use these steps:
请使用以下步骤：
- List the first few terms from the problem | 列出问题中的前几项
- Find the difference between consecutive terms | 求相邻项之间的差
- Write the position-to-term rule | 写出位置与项的关系规则
- Substitute n to answer the question | 代入 n 回答问题
This method works for many IGCSE questions involving savings, seats, steps, and tile patterns.
这种方法适用于许多 IGCSE 题目，包括储蓄、座位、台阶和瓷砖规律问题。
10. Exam-Style Practice and Common Mistakes | 考试风格练习与常见错误
Practice question: Find the nth term of 2, 9, 16, 23, 30 ... and use it to find the 100th term.
练习题：求数列 2、9、16、23、30…… 的第 n 项，并用它求第 100 项。
The common difference is 7 and the first term is 2, so aₙ = 2 + (n − 1) × 7 = 7n − 5.
公差为 7，首项为 2，所以 aₙ = 2 + (n − 1) × 7 = 7n − 5。
Therefore a₁₀₀ = 7 × 100 − 5 = 695.
因此 a₁₀₀ = 7 × 100 − 5 = 695。
A common mistake is to use the common difference as the whole formula. For example, writing "7n" instead of "7n − 5" ignores the correction needed for the first term.
一个常见错误是把公差直接当成完整公式。例如写成 7n 而不是 7n − 5，忽略了首项所需的修正。
Always check your nth term by substituting n = 1. It must give the first term exactly.
始终通过代入 n = 1 来检查你的第 n 项公式，它必须恰好得到首项。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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