A-Level Maths: Arithmetic Sequences — Concepts and the General Term | A-Level 数学：等差数列的概念与通项公式

📚 A-Level Maths: Arithmetic Sequences — Concepts and the General Term | A-Level 数学：等差数列的概念与通项公式

An arithmetic sequence is one of the most fundamental topics in A-Level Mathematics. It appears not only in Pure Mathematics but also serves as a foundation for series, summations, and even modelling in statistics and mechanics. Understanding its definition and the general term formula is essential for success in exams.

等差数列是 A-Level 数学中最基础的主题之一。它不仅出现在纯数学中，还是级数、求和乃至统计与力学建模的基石。理解其定义与通项公式，对考试取得好成绩至关重要。

1. What Is an Arithmetic Sequence? | 什么是等差数列？

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant is called the common difference, usually denoted by the letter d.

等差数列是一列数，其中任意两个相邻项之间的差是常数。这个常数称为公差，通常用字母 d 表示。

For example, the sequence 3, 7, 11, 15, 19 is an arithmetic sequence because each term increases by 4. Here, the first term a = 3 and the common difference d = 4.

例如，数列 3, 7, 11, 15, 19 是等差数列，因为每一项增加 4。这里首项 a = 3，公差 d = 4。

2. The General Term Formula | 通项公式

The n-th term of an arithmetic sequence is given by the formula:

等差数列的第 n 项由以下公式给出：

uₙ = a + (n − 1)d

where uₙ is the n-th term, a is the first term, d is the common difference, and n is the term number (a positive integer).

其中 uₙ 是第 n 项，a 是首项，d 是公差，n 是项数（正整数）。

3. Deriving the General Term | 推导通项公式

To derive the formula, observe that the terms of the sequence can be written as:

为了推导这一公式，我们可以将数列的各项写成：

u₁ = a, u₂ = a + d, u₃ = a + 2d, u₄ = a + 3d, …

Each term adds one more d than the previous one. By the time we reach the n-th term, we have added d exactly (n − 1) times. Hence, uₙ = a + (n − 1)d.

每一项比前一项多一个 d。当我们到达第 n 项时，已经加了 (n − 1) 个 d。因此 uₙ = a + (n − 1)d。

4. Worked Example 1: Finding the 20th Term | 例题 1：求第 20 项

Given the arithmetic sequence 5, 9, 13, 17, …, find the 20th term.

已知等差数列 5, 9, 13, 17, …，求第 20 项。

Here, a = 5 and d = 4. Using the formula:

这里 a = 5，d = 4。使用公式：

u₂₀ = 5 + (20 − 1) × 4 = 5 + 76 = 81

Therefore, the 20th term is 81.

因此，第 20 项是 81。

5. Worked Example 2: Finding the First Term | 例题 2：求首项

The 8th term of an arithmetic sequence is 33 and the common difference is 4. Find the first term.

某等差数列的第 8 项为 33，公差为 4。求首项。

Using uₙ = a + (n − 1)d, substitute n = 8, u₈ = 33, d = 4:

利用 uₙ = a + (n − 1)d，代入 n = 8，u₈ = 33，d = 4：

33 = a + (8 − 1) × 4 = a + 28 ⇒ a = 5

Thus the first term is 5 and the full sequence starts 5, 9, 13, 17, 21, 25, 29, 33.

因此首项为 5，完整数列从 5, 9, 13, 17, 21, 25, 29, 33 开始。

6. Worked Example 3: Finding the Common Difference | 例题 3：求公差

The 6th term of an arithmetic sequence is 26 and the 12th term is 50. Find the common difference.

某等差数列的第 6 项是 26，第 12 项是 50。求公差。

Write two equations using the formula:

用公式写出两个方程：

u₆ = a + 5d = 26
u₁₂ = a + 11d = 50

Subtract the first equation from the second:

用第二个方程减去第一个方程：

(a + 11d) − (a + 5d) = 50 − 26 ⇒ 6d = 24 ⇒ d = 4

The common difference is 4. We can then substitute back to find a = 6 if needed.

公差为 4。如有需要，可代回求出 a = 6。

7. The Common Difference Can Be Negative | 公差可以为负

An arithmetic sequence does not have to be increasing. If d < 0, the sequence decreases. For example, 20, 16, 12, 8, 4 is an arithmetic sequence with a = 20 and d = −4.

等差数列不一定递增。若 d < 0，数列递减。例如，20, 16, 12, 8, 4 是一个等差数列，其中 a = 20，d = −4。

The general term formula works for negative differences without any modification:

通项公式对负公差同样适用，无需任何修改：

uₙ = 20 + (n − 1) × (−4) = 24 − 4n

8. Zero in Arithmetic Sequences | 等差数列中的零项

In some problems, you may be asked whether a particular number appears in a sequence. This involves setting uₙ equal to the target value and solving for n.

有些题目会问某个数是否为数列中的一项。这需要将 uₙ 设为目标值，然后解出 n。

For example, given uₙ = 3n + 2, does 100 appear in the sequence? Set 3n + 2 = 100, giving 3n = 98, so n = 32.67. Since n is not an integer, 100 is not a term.

例如，给定 uₙ = 3n + 2，问 100 是否出现在数列中？令 3n + 2 = 100，得 3n = 98，所以 n = 32.67。由于 n 不是整数，100 不是数列中的一项。

9. Graphical Interpretation | 图形的解读

When plotted against n, the terms of an arithmetic sequence lie on a straight line. The gradient of this line equals the common difference d, and the vertical intercept is a − d.

将 n 作为横轴绘图时，等差数列的项落在一条直线上。这条直线的斜率等于公差 d，纵截距为 a − d。

This is because uₙ = dn + (a − d), which is a linear function of n. This connection helps you solve geometry-related sequence questions quickly.

这是因为 uₙ = dn + (a − d)，它是 n 的线性函数。理解这一联系有助于快速解决与几何相关的数列问题。

10. Common Exam Pitfalls | 常见考试陷阱

Using uₙ = a + nd instead of a + (n − 1)d. Remember: the first term has zero added differences, so for n = 1 the formula must return a.
混淆 uₙ = a + nd 与 a + (n − 1)d。记住：首项不添加任何公差，因此当 n = 1 时公式必须返回 a。
Forgetting that n must be a positive integer when checking whether a value belongs to the sequence.
忘记检查 n 是否为正整数——判断某值是否为数列中的项时必须验证。
Mixing up the common difference d with the common ratio r used in geometric sequences.
将等差中的公差 d 与等比中的公比 r 混淆。

11. Connection to Arithmetic Series | 与等差数列前 n 项和的联系

The sum of the first n terms of an arithmetic sequence is given by:

等差数列前 n 项的和由下式给出：

Sₙ = n⁄2 [2a + (n − 1)d] = n⁄2 (a + uₙ)

This formula relies on the general term uₙ, so mastering the general term is necessary before tackling sums. Many exam questions first ask for uₙ and then for Sₙ.

该公式依赖通项 uₙ，因此在处理求和之前，必须掌握通项公式。许多考题先要求 uₙ，再求 Sₙ。

12. Summary and Final Advice | 总结与考前建议

The arithmetic sequence is defined by its first term a and common difference d. The general term uₙ = a + (n − 1)d is the single most important formula in this topic. Practice identifying a and d correctly, and always check your answer by substituting a small value of n.

等差数列由首项 a 和公差 d 定义。通项公式 uₙ = a + (n − 1)d 是本主题中最核心的公式。练习时务必正确识别 a 和 d，并通过代入小的 n 值检查结果。

As you attempt past papers, write down the formula at the start of every sequence problem. This reduces careless errors and makes your method clear to the examiner.

在练习历年试卷时，建议在每道数列题的开头写下通项公式。这样可以减少粗心错误，也让阅卷者看清你的解题思路。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

A-Level Maths: Arithmetic Sequences — Concepts and the General Term | A-Level 数学：等差数列的概念与通项公式