📚 Number Patterns and Sequences | 数字规律与数列
In mathematics, a sequence is an ordered list of numbers. Each number in the list is called a term. Learning to recognise number patterns helps us predict the next term, solve real-life problems, and prepare for more advanced algebra.
在数学中，数列是按一定顺序排列的一组数。其中的每一个数都称为“项”。学会识别数字规律，可以帮助我们预测下一项、解决实际问题，并为更高阶的代数学习打下基础。
1. What Are Number Patterns? | 什么是数字规律
A number pattern is a sequence that follows a rule. The rule tells us how to move from one term to the next term. A sequence can increase, decrease, or even alternate between different operations.
数字规律是指遵循某种规则的数列。规则告诉我们如何从一个项得到下一个项。数列可以递增、递减，甚至交替使用不同的运算。
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Example 1: 2, 4, 6, 8, … The rule is “add 2”.
例1：2, 4, 6, 8, … 规律是“每次加2”。
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Example 2: 3, 6, 12, 24, … The rule is “multiply by 2”.
例2：3, 6, 12, 24, … 规律是“每次乘以2”。
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Example 3: 700, 690, 680, 670, … The rule is “subtract 10”.
例3：700, 690, 680, 670, … 规律是“每次减10”。
| Pattern | Rule | Next term |
| 2, 4, 6, 8 | +2 | 10 |
| 3, 6, 12, 24 | ×2 | 48 |
| 700, 690, 680, 670 | -10 | 660 |
2. Arithmetic Sequences | 等差数列
An arithmetic sequence is a sequence where the difference between consecutive terms is always the same. This constant difference is called d. The word “arithmetic” here means that we add or subtract the same number each time.
等差数列是指相邻两项之差始终相同的数列。这个固定的差称为 d。这里的“等差”意味着我们每次都加上或减去同一个数。
For example, the sequence 700, 720, 740, 760, … is arithmetic because each term increases by 20.
例如，数列 700, 720, 740, 760, … 是等差数列，因为每一项增加20。
aₙ = a₁ + (n – 1)d
In this formula, aₙ is the nth term, a₁ is the first term, n is the term number, and d is the common difference.
在这个公式中，aₙ 是第 n 项，a₁ 是第一项，n 是项数，d 是公差。
If a₁ = 700 and d = 15, then the 10th term is a₁₀ = 700 + (10 – 1) × 15 = 700 + 135 = 835.
如果 a₁ = 700，d = 15，那么第10项是 a₁₀ = 700 + (10 – 1) × 15 = 700 + 135 = 835。
3. Geometric Sequences | 等比数列
A geometric sequence is a sequence where each term is multiplied by the same number to get the next term. This number is called the common ratio, r.
等比数列是指每一项都乘以同一个数从而得到下一项的数列。这个数称为公比，记作 r。
For example, 2, 6, 18, 54, … is geometric because each term is multiplied by 3.
例如，2, 6, 18, 54, … 是等比数列，因为每一项都乘以3。
aₙ = a₁ × rⁿ⁻¹
Here, aₙ is the nth term, a₁ is the first term, r is the common ratio, and n is the term number.
其中，aₙ 是第 n 项，a₁ 是第一项，r 是公比，n 是项数。
For the sequence 2, 6, 18, 54, … we have a₁ = 2 and r = 3. The 6th term is 2 × 3⁵ = 2 × 243 = 486.
对于数列 2, 6, 18, 54, …，a₁ = 2，r = 3。第6项是 2 × 3⁵ = 2 × 243 = 486。
Geometric patterns also appear in everyday life, such as doubling money or the growth of bacteria.
等比规律也出现在日常生活中，例如金钱翻倍或细菌繁殖。
4. Square, Cube and Triangular Numbers | 平方数、立方数与三角形数
Square numbers are formed by multiplying a whole number by itself: 1, 4, 9, 16, 25, … They can be written as 1², 2², 3², 4², 5², …
平方数是由一个整数自乘得到的：1, 4, 9, 16, 25, … 它们也可以写作 1², 2², 3², 4², 5², …
Cube numbers are formed by multiplying a whole number by itself twice: 1, 8, 27, 64, 125, …
立方数是由一个整数连乘自身三次得到的：1, 8, 27, 64, 125, …
Tth triangular numbers come from arranging dots in a triangle. The sequence starts 1, 3, 6, 10, 15, …
三角形数可以看作按三角形排列的点数。数列从 1, 3, 6, 10, 15, … 开始。
The formula for the nth triangular number is:
第 n 个三角形数的公式为：
Tₙ = n(n + 1) ÷ 2
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Example: The 7th square number is 7² = 49.
例：第7个平方数是 7² = 49。
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Example: The 5th cube number is 5³ = 125.
例：第5个立方数是 5³ = 125。
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Example: The 4th triangular number is T₄ = 4 × 5 ÷ 2 = 10.
例：第4个三角形数是 T₄ = 4 × 5 ÷ 2 = 10。
5. Fibonacci-Type Sequences | 斐波那契型数列
A Fibonacci-type sequence is made by adding the previous two terms to get the next term. The most famous Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, …
斐波那契型数列的规则是：前两项之和等于下一项。最著名的斐波那契数列是 1, 1, 2, 3, 5, 8, 13, …
You can start with any two numbers. For example, if we start with 700 and 200, the next term is 700 + 200 = 900, and the term after that is 200 + 900 = 1100.
你可以用任意两个数开始。例如，如果我们从 700 和 200 开始，下一项就是 700 + 200 = 900，再下一项是 200 + 900 = 1100。
Sequence: 700, 200, 900, 1100, 2000, 3100, …
Fibonacci-type patterns appear in nature, for example in the arrangement of leaves and petals.
斐波那契型规律也出现在自然界中，例如叶子和花瓣的排列。
6. Finding the nth Term | 求第 n 项
For a linear sequence, we can find an expression for the nth term. A linear sequence has a constant difference, and its nth term has the form d × n + c.
对于线性数列，我们可以求出第 n 项的表达式。线性数列有固定公差，其第 n 项具有 d × n + c 的形式。
Step 1: Find the common difference d.
第一步：找到公差 d。
Step 2: Compare the sequence with d × n.
第二步：把数列与 d × n 进行比较。
Step 3: Add or subtract a constant c to match the first term.
第三步：加上或减去一个常数 c，使结果与第一项一致。
Consider the sequence 5, 9, 13, 17, … The difference is 4, so start with 4n. For n = 1, 4n = 4, but the first term is 5, so add 1. Therefore the nth term is 4n + 1.
观察数列 5, 9, 13, 17, …。公差是4，所以先写 4n。当 n = 1 时，4n = 4，但第一项是5，所以要加1。因此第 n 项是 4n + 1。
Check this formula: for n = 3, 4 × 3 + 1 = 13, which matches the third term.
验证这个公式：当 n = 3 时，4 × 3 + 1 = 13，正好等于第三项。
Now consider 700, 684, 668, 652, … The difference is -16. Start with -16n. For n = 1, -16n = -16, but the first term is 700. We need to add 716, so the nth term is 716 – 16n.
再看 700, 684, 668, 652, …。公差是 -16。先写 -16n。当 n = 1 时，-16n = -16，但第一项是700。我们需要加上716，所以第 n 项是 716 – 16n。
7. Pattern Tables and Diagrams | 图表中的规律
Many mathematical problems show patterns using diagrams. Drawing a table is a powerful way to organise the information and find the rule.
许多数学问题会通过图形展示规律。画一个表格是整理信息并找到规则的强大方法。
Look at a pattern made from matchstick squares. One square needs 4 sticks, two squares need 7 sticks, and three squares need 10 sticks.
观察一个用火柴棒拼正方形的图形。拼1个正方形需要4根火柴棒，拼2个需要7根，拼3个需要10根。
| Number of squares n | 1 | 2 | 3 | 4 |
| Number of sticks | 4 | 7 | 10 | 13 |
The differences are always 3, so the nth term is 3n + 1. This means 10 squares use 3 × 10 + 1 = 31 sticks.
每一项的差始终是3，所以第 n 项是 3n + 1。这说明拼10个正方形需要 3 × 10 + 1 = 31 根火柴棒。
Always check a formula using more than one term from the table.
使用公式时，一定要用表格中的多个项进行验证。
8. Special Sequences: Primes and Multiples | 特殊数列：质数与倍数
Prime numbers are numbers greater than 1 that have exactly two factors: 1 and themselves. The first few primes are 2, 3, 5, 7, 11, 13, …
质数是大于1且只有两个因数的数，这两个因数是1和它本身。前几个质数是 2, 3, 5, 7, 11, 13, …
Multiples of a number are made by multiplying that number by whole numbers. For example, the multiples of 7 are 7, 14, 21, 28, 35, …
一个数的倍数是由这个数乘以整数得到的。例如，7的倍数是 7, 14, 21, 28, 35, …
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700 is a multiple of 7 because 700 = 7 × 100.
700 是 7 的倍数，因为 700 = 7 × 100。
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700 is also a multiple of 100 because 700 = 100 × 7.
700 也是 100 的倍数，因为 700 = 100 × 7。
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700 is an even number, so its last digit is 0.
700 是偶数，所以它的末位数字是0。
Other special sequences include even numbers 2, 4, 6, 8, … and odd numbers 1, 3, 5, 7, …
其他特殊数列包括偶数数列 2, 4, 6, 8, … 和奇数数列 1, 3, 5, 7, …
9. Common Mistakes | 常见错误
When studying number patterns, students often make several common mistakes. Knowing these will help you avoid them in exams.
在学习数字规律时，学生常常会犯一些常见错误。了解这些错误可以帮助你在考试中避免它们。
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Mistake 1: Confusing arithmetic and geometric sequences. Arithmetic uses addition or subtraction, while geometric uses multiplication or division.
错误一：混淆等差数列与等比数列。等差数列用加法或减法，等比数列用乘法或除法。
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Mistake 2: Assuming a pattern is arithmetic when it is actually quadratic or Fibonacci-type.
错误二：当规律实际上是二次型或斐波那契型时，却误以为它是等差数列。
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Mistake 3: Forgetting to check the nth term formula with more than one term.
错误三：只用一项验证第 n 项公式，忘记用多个项检查。
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Mistake 4: Misreading negative differences. In 700, 690, 680, … the difference is -10, not +10.
错误四：读错负公差。在 700, 690, 680, … 中，公差是 -10，不是 +10。
10. Practice Questions | 练习题目
Try these questions by yourself before reading the answers.
先自己尝试完成以下题目，再看答案。
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Question 1: Find the next term in the sequence 3, 7, 11, 15, …
题目1：求数列 3, 7, 11, 15, … 的下一项。
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Answer 1: 19, because the common difference is 4.
答案1：19，因为公差是4。
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Question 2: Find the 6th term of the sequence 10, 20, 40, 80, …
题目2：求数列 10, 20, 40, 80, … 的第6项。
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