📚 Arithmetic and Geometric Sequences | 等差与等比数列
Sequences are one of the fundamental building blocks of algebra and appear extensively in the Edexcel IGCSE Mathematics syllabus. Being able to recognise whether a sequence is arithmetic or geometric, and to work confidently with the related formulas for the nth term and sums of series, is essential for success in both Foundation and Higher tier exams. This article will guide you through definitions, key formulas, worked examples, and common pitfalls to avoid.
数列是代数的基础模块之一，在 Edexcel IGCSE 数学大纲中频繁出现。能否判断一个数列是等差数列还是等比数列，并熟练运用第 n 项公式与求和公式，对于在基础卷和进阶卷中取得好成绩至关重要。本文将带你梳理定义、关键公式、典型例题以及需要避免的常见错误。
1. Definitions of Sequences | 数列的定义
A sequence is simply an ordered list of numbers, each called a term. We usually denote the first term as u1 or a, the second term as u2, and the nth term as un. Sequences can be finite (having a limited number of terms) or infinite. The two most important types encountered at IGCSE are arithmetic and geometric sequences.
数列就是一组按顺序排列的数字，每个数字称为一项。我们通常将第一项记作 u1 或 a，第二项记作 u2，第 n 项记作 un。数列可以是有限的（项数有限）也可以是无限的。IGCSE 阶段最重要的两类数列是等差数列和等比数列。
2. Arithmetic Sequences: Common Difference | 等差数列：公差
h the difference between consecutive terms is constant. This constant is called the common difference, usually written as d. For example, in the sequence 4, 7, 10, 13, … 3. To check whether a sequence is arithmetic, subtract any term from the following term: un+1 – un must always give the same value.
等差数列是相邻两项的差始终相等的数列。这个不变的差称为公差，通常记作 d。例如数列 4, 7, 10, 13, … 的公差 d = 3。要判断一个数列是否为等差数列，只需用后一项减去前一项：如果 un+1 – un 的值始终相同，则数列为等差数列。
3. nth Term of an Arithmetic Sequence | 等差数列的第 n 项
The formula for the nth term of an arithmetic sequence is: un = a + (n – to find any term in the sequence without listing them all. For the sequence 4, 7, 10, 13, …, the 50th term is u50 = 4 + (50 – 1)×3 = 4 + 49×3 = 151.
等差数列第 n 项的公式为：un = a + (n – 1)d，其中 a 为首项，d 为公差，n 为项数。这个线性公式能让你直接求出数列中的任意一项，而无需逐一列出。对于数列 4, 7, 10, 13, …，第 50 项为 u50 = 4 + (50 – 1)×3 = 4 + 49×3 = 151。
4. Sum of an Arithmetic Series | 等差数列的求和
When we add the terms of an arithmetic sequence we obtain an arithmetic series. The sum of the first n terms, denoted Sn, can be calculated using two equivalent forms: Sn = n/2 [2a + (n – 1)d] or Sn he second version is especially useful when you know the first and last terms. For example, the sum of the first 20 terms of 4, 7, 10, … is S20 = 20/2 [2×4 + (20 – 1)×3] = 10 × [8 + 57] = 650.
将等差数列的各项相加就得到了等差数列的求和。前 n 项和记作 Sn，可用两个等价公式计算：Sn = n/2 [2a + (n – 1)d] 或 Sn = n/2 (a + l)，其中 l 为末项。第二个公式在已知首项和末项时特别有用。例如，数列 4, 7, 10, … 的前 20 项和为 S20 = 20/2 [2×4 + (20 – 1)×3] = 10 × [8 + 57] = 650。
5. Geometric Sequences: Common Ratio | 等比数列：公比
onstant is called the common ratio, denoted by r. For instance, the sequence 3, 6, 12, 24, … has r = 2. You can find r by dividing a term by its predecessor: un+1 / un must equal r for all n. If |r| < 1, the terms get closer to zero; if |r| > 1, the terms grow rapidly.
等比数列是指每一项与前一项的比值为常数的数列。这个常数称为公比，记作 r。例如数列 3, 6, 12, 24, … 的公比 r = 2。求公比的方法是用一项除以前一项：对所有 n，un+1 / un 必须等于 r。如果 |r| < 1，数列各项会逐渐趋近于零；若 |r| > 1，则各项快速增大。
6. nth Term of a Geometric Sequence | 等比数列的第 n 项
The nth term of a geometric sequence is given by: un = a rn–1, where a is the first term, r is the common ratio, and n is the term position. This exponential formula is used to generate any term directly. For the sequence 3, 6, 12, 24, …, the 8th term is u8 = 3 × 28–1 = 3 × 27 = 3 × 128 = 384. Remember that the exponent is n–1, not n.
等比数列的第 n 项公式为：un = a rn–1，其中 a 为首项，r 为公比，n 为项的位置。通过这个指数型公式可以直接求出任意一项。对于数列 3, 6, 12, 24, …，第 8 项为 u8 = 3 × 28–1 = 3 × 27 = 3 × 128 = 384。务必记住指数是 n–1，而不是 n。
7. Sum of a Finite Geometric Series | 有限等比数列的求和
The sum of the first n terms of a geometric series is Sn = a(rn – 1) / (r – 1), valid for r ≠ 1. If r < 1, the formula is often rewritten as Sn = a(1 – rn) / (1 – r) to avoid negative numerator. For the series 3 + 6 + 12 + ... up to 8 terms, S8 = 3(28 – 1) / (2 – 1) = 3(256 – ing problems involving compound interest, savings schemes, or repeated percentage changes.
有限等比数列前 n 项和的公式为 Sn = a(rn – 1) / (r – 1)，适用于 r ≠ 1。当 r < 1 时，常改写为 Sn = a(1 – rn) / (1 – r) 以避免分子出现负号。对于级数 3 + 6 + 12 + ... 前 8 项，S8 = 3(28 – 1) / (2 – 1) = 3(256 – 1) = 765。这一公式在解决复利、储蓄计划或重复百分比变化的问题中必不可少。
8. Sum to Infinity of a Geometric Series | 无限等比数列的求和
When the common ratio satisfies |r| < 1, an infinite geometric series converges to a finite sum. The sum to infinity is given by S∞ = a / (1 – r). For example, the series 8 + 4 + 2 + 1 + … has a = 8, r = ½, so S∞ = 8 / (1 – This concept often appears in questions about recurring decimals or perpetual patterns. If |r| ≥ 1, the series diverges and no finite sum exists.
当公比满足 |r| < 1 时，无限等比数列会收敛于一个有限的和。无穷和公式为 S∞ = a / (1 – r)。例如，级数 8 + 4 + 2 + 1 + … 中 a = 8，r = ½，因此 S∞ = 8 / (1 – ½) = 16。这一概念常出现在循环小数或无限模式的题目中。如果 |r| ≥ 1，则该级数发散，不存在有限和。
9. Recognising Sequences in Context | 根据情境识别数列
Exam questions may describe a scenario that hides a sequence. Keywords like “adds the same amount each week” suggest an arithmetic sequence, while “multiplies by a fixed percentage” or “doubles every hour” point to a geometric sequence. Practise translating word problems into a, n, d or r, then apply the correct formulas. Making a small table of the first few terms can help you deduce the pattern.
考试题目有时会用一个具体情境来隐藏数列。像“每周增加相同数量”这样的描述暗示是等差数列，而“按固定百分比增长”或“每小时翻倍”则指向等比数列。学会将文字题转化为 a, n, d 或 r，然后套用正确的公式。先写出前几项的简易表格有助于推断出规律。
10. Common Mistakes and Tips | 常见错误与技巧
One frequent error is using n instead of n–1 in the nth term formulas. Always remember: for arithmetic, un = a + (n – 1)d, and for geometric, un = a rn–1. Another mistake is misidentifying the common difference or ratio; double-check by comparing several consecutive pairs. When summing geometric series, ensure you use the correct bracket placement in your calculator. Also, confirm whether the problem asks for the sum of terms or the nth term – these are easily confused under exam pressure.
一个常见错误是在第 n 项公式中误用 n 而不是 n–1。一定要记住：等差数列 un = a + (n – 1)d，等比数列 un = a rn–1。另一个错误是辨错公差或公比；应通过比较多组相邻项来复核。在等比数列求和时，务必在计算器上正确输入括号。还需确认题目要求的是各项之和还是第 n 项——在考试压力下这两者很容易混淆。
11. Practice Questions | 练习题举例
To solidify your understanding, try these typical Edexcel IGCSE style questions: (i) Find the 15th term of the arithmetic sequence 5, 12, 19, … (ii) The 3rd term of an arithmetic sequence is 10 and the 8th term is 25. Determine a and d. (iii) Calculate the sum of the first 12 terms of the geometric series starting 2 + 6 + 18 + … (iv) A ball bounces to 80% of its previous height each time; the first bounce reaches 200 cm. Find the total vertical distance travelled by the ball before it comes to rest.
为巩固理解，不妨尝试以下典型的 Edexcel IGCSE 风格题目：(i) 求等差数列 5, 12, 19, … 的第 15 项。(ii) 若等差数列的第 3 项为 10，第 8 项为 25，求 a 和 d。(iii) 计算以 2 + 6 + 18 + … 开头的等比级数的前 12 项和。(iv) 一个皮球每次弹起的高度是前一次的 80%；第一次弹起高度为 200 cm。求球在静止前经过的总垂直距离。
12. Summary and Key Formula Sheet | 总结与核心公式表
Always keep this summary handy when revising. For an arithmetic sequence: un = a + (n – 1)d, Sn = n/2 [2a + (n – 1)d] = n/2 (a + l). For a geometric sequence: un = a rn–1, Sn = a(rn – 1)/(r – 1) for r > 1, or Sn = a(1 – rn)/(1 – r) for r < 1, and S∞ = a/(1 – r) provided |r| < 1. Mastering these will give you a solid foundation for tackling sequences questions.
复习时请将以下总结放在手边。等差数列：un = a + (n – 1)d，Sn = n/2 [2a + (n – 1)d] = n/2 (a + l)。等比数列：un = a rn–1，当 r > 1 时 Sn = a(rn – 1)/(r – 1)，当 r < 1 时 Sn = a(1 – rn)/(1 – r)，且当 |r| < 1 时 S∞ = a/(1 – r)。掌握这些公式将为你解答数列相关题目奠定坚实的基础。
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