📚 Arithmetic Sequences: Finding the nth Term | 等差数列：求第 n 项
In IGCSE Mathematics, sequences are ordered lists of numbers that follow a specific rule. Arithmetic sequences – also called linear sequences – are built by adding (or subtracting) the same value each step, making them the most predictable type of sequence. Mastering the skill of finding the nth term expression turns a simple pattern into a powerful tool that can predict any term instantly, saving time in exams and building a strong foundation for more advanced algebra.
在 IGCSE 数学中，数列是按特定规则排列的一串数字。等差数列（也叫线性数列）每一步都加上（或减去）同一个数值，因此是最容易预测的数列类型。掌握求第 n 项表达式的技巧，可以把一个简单的规律转化为预知任意项的强大工具，考试时省时高效，也为更高级的代数打下坚实基础。
1. What is a Sequence? | 什么是数列？
A sequence is a set of numbers written in a definite order. Each number is called a term, and terms are often labelled T₁, T₂, T₃, … where the subscript tells you the position of the term. In Edexcel IGCSE, you will mostly work with linear sequences where the difference between consecutive terms stays the same – these are arithmetic sequences.
数列是按照确定顺序排列的一列数。每个数称为一项，项常用 T₁, T₂, T₃, … 表示，下标代表该项所在的位置。在 Edexcel IGCSE 数学中，你接触的多数是线性数列，也就是相邻项的差恒定不变——这类数列就是等差数列。
2. Terms and Position Numbers | 项与项数
The position number n tells you the order of a term: n = 1 for the first term, n = 2 for the second term, and so on. When you describe a sequence, always connect the term value with its position. For example, in the sequence 4, 7, 10, 13, … the first term (n = 1) is 4, the second term is 7, the third is 10. This relationship can be expressed as a formula in terms of n.
项数 n 表明了项的顺序：n = 1 对应首项，n = 2 对应第二项，以此类推。描述数列时，务必把项的值和它的位置联系起来。例如，数列 4, 7, 10, 13, … 中，首项（n = 1）是 4，第二项是 7，第三项是 10。这种关系最终可以用含有 n 的公式表示。
3. Common Difference in Linear Sequences | 线性数列的公差
The common difference d is the amount added (or subtracted) from one term to the next. To find d, subtract any term from the term that follows it: d = T₂ − T₁ = T₃ − T₂, and so on. If the difference is positive, the sequence increases; if negative, it decreases. In the linear sequence 5, 9, 13, 17, … the common difference is +4.
公差 d 是从某一项到下一项所加（或减）的固定数值。求 d 时，用后一项减去前一项：d = T₂ − T₁ = T₃ − T₂，依次类推。若差值为正，数列递增；若为负，数列递减。在数列 5, 9, 13, 17, … 中，公差为 +4。
d = Tₙ₊₁ − Tₙ
4. The nth Term Formula | 第 n 项公式
For any arithmetic sequence, the nth term Tₙ can be found using the rule Tₙ = a + (n − 1)d, where a is the first term and d is the common difference. This formula works because you start at a and add d repeatedly. In Edexcel IGCSE, you are often asked to write the nth term as a simplified expression like 3n + 1 or 7 − 2n.
对于任意等差数列，第 n 项 Tₙ 都可以用规则 Tₙ = a + (n − 1)d 求得，其中 a 为首项，d 为公差。该公式之所以成立，是因为你从 a 出发，每次都加上 d。在 Edexcel IGCSE 考试中，经常要求写出诸如 3n + 1 或 7 − 2n 这样化简后的第 n 项表达式。
Tₙ = a + (n − 1)d
5. Finding the Common Difference from a Sequence | 从数列中求公差
Look at the first few terms and find the jump between consecutive terms. If the sequence is 8, 5, 2, −1, …, subtracting gives 5 − 8 = −3, 2 − 5 = −3, so d = −3. The sequence is decreasing. Once you identify d, you can quickly check it stays the same throughout the given terms.
观察前几项，求出相邻项之间的跳跃量。若数列为 8, 5, 2, −1, …, 用后项减前项得 5 − 8 = −3，2 − 5 = −3，因此 d = −3，数列是递减的。确定 d 后，你可以迅速验证它在所有已知项中是否保持不变。
6. Finding the Zeroth Term (a − d) | 求第零项 (a − d)
A useful trick in linear sequences is to imagine the term before the first term, called the zeroth term. Its value is a − d. This value matches the constant part in the nth term expression when written as dn + c. For instance, if T₁ = 4 and d = 3, the zeroth term is 4 − 3 = 1. The nth term will be 3n + 1, where 1 is the constant.
线性数列中一个实用的技巧是想象出第一项之前的那一项，称为第零项。它的值为 a − d。这个值与第 n 项写成 dn + c 时的常数部分相对应。例如，若 T₁ = 4 且 d = 3，第零项就是 4 − 3 = 1，那么第 n 项就是 3n + 1，其中 1 就是常数项。
Zeroth term = First term − Common difference
7. Writing the nth Term Expression | 写出第 n 项表达式
Combine the common difference d and the constant c obtained from the zeroth term. Write the expression in the form dn + c. Always simplify: if d = 2 and the zeroth term = 5, the nth term is 2n + 5. If the sequence decreases, d will be negative, e.g. −3n + 14. Check by substituting n = 1, you should get the first term.
将公差 d 与从第零项得到的常数 c 组合起来，写成 dn + c 的形式。务必化简：若 d = 2 且第零项 = 5，第 n 项就是 2n + 5。若数列递减，d 为负数，例如 −3n + 14。为了验证，代入 n = 1 应得到首项。
8. Using the nth Term to Find Any Term | 使用第 n 项求任意项
Once you have the expression, substitute the desired position number for n. To find the 20th term of the sequence 4, 9, 14, 19, … first identify d = 5 and the zeroth term: 4 − 5 = −1, so the nth term is 5n − 1. Then T₂₀ = 5(20) − 1 = 100 − 1 = 99. This method prevents having to list all terms up to the 20th.
得出表达式后，将想要的位置数代入 n。例如，要求数列 4, 9, 14, 19, … 的第 20 项，首先确定 d = 5，第零项为 4 − 5 = −1，因此第 n 项为 5n − 1。那么 T₂₀ = 5(20) − 1 = 100 − 1 = 99。这个方法使我们不必列出前 20 项。
| n | 5n − 1 | Term |
|---|---|---|
| 1 | 5(1)−1 | 4 |
| 2 | 5(2)−1 | 9 |
| 3 | 5(3)−1 | 14 |
| 10 | 5(10)−1 | 49 |
| 20 | 5(20)−1 | 99 |
9. Finding the nth Term from Given Terms | 由给定项求第 n 项
Sometimes you are given two terms, for example the 3rd term is 10 and the 7th term is 26. The difference in position is 7 − 3 = 4, and the difference in value is 26 − 10 = 16, so d = 16 ÷ 4 = 4. Then use Tₙ = a + (n−1)d with one term to find a: 10 = a + (3−1)×4 → a = 10 − 8 = 2. The nth term is 2 + (n−1)×4 = 4n − 2.
有时题目给出两个项，比如第 3 项是 10，第 7 项是 26。位置差为 7 − 3 = 4，值的差为 26 − 10 = 16，因此 d = 16 ÷ 4 = 4。接着用 Tₙ = a + (n−1)d 代入其中一项求 a：10 = a + (3−1)×4 → a = 10 − 8 = 2。于是第 n 项为 2 + (n−1)×4 = 4n − 2。
d = (Value difference) ÷ (Position difference)
10. Real‑Life Applications and Exam Tips | 实际应用与考试技巧
Arithmetic sequences appear everywhere: saving a fixed amount each week, arranging seats in rows, or predicting scores. In Edexcel IGCSE, questions may ask you to decide if a given number belongs to the sequence. Solve dn + c = that number for n. If n is a positive integer, it is a term. Always double‑check your nth term by testing n = 1, n = 2 and n = 3. Write the final expression clearly, and remember to put a negative sign if d is negative.
等差数列在生活中比比皆是：每周固定存钱、安排成排的座位，或预测得分。在 Edexcel IGCSE 中，题目可能要求判断某个数是否属于该数列。方法是解方程 dn + c = 那个数，若解出的 n 是正整数，它就是其中的一项。始终要用 n = 1, n = 2 和 n = 3 检验你的第 n 项。书写最终表达式时要清晰，如果 d 为负，千万别漏掉负号。
- Check the common difference twice to avoid sign errors. / 检查公差两次，避免符号错误。
- Use brackets when substituting to prevent careless mistakes. / 代入时使用括号，防止粗心错误。
- Present the nth term in simplest form, e.g. 2n − 5, not 2n + (−5). / 将第 n 项化为最简形式，如 2n − 5 而不是 2n + (−5)。
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