📚 Finding the nth Term of a Sequence | 寻找数列的第n项
Sequences are ordered lists of numbers that follow a rule. In KS3 Cambridge Mathematics, learning to find the nth term of a sequence is essential for understanding patterns and preparing for algebra. This article revisits the key skills found in a typical practice exercise (such as Page 41, Exercise 1) and provides clear explanations and examples to build your confidence.
数列是按照一定规则排列的一列数。在剑桥KS3数学中，学会寻找数列的第n项对于理解模式和代数至关重要。本文重温了典型练习题（例如第41页练习1）中的关键技能，并通过清晰的解释和示例来增强你的信心。
1. What is a Sequence? | 什么是数列？
A sequence is an ordered set of numbers, called terms, that follow a specific pattern or rule. Each term can be worked out from its position. For example, 3, 6, 9, 12, ... is a sequence where each term increases by 3. The first term is 3, the second is 6, and so on.
数列是一组有序的数，每一项都遵循特定的模式或规则。每项都可以根据其位置推算出来。例如，3, 6, 9, 12, ... 是一个数列，每一项增加3。第一项是3，第二项是6，以此类推。
2. Linear Sequences | 线性数列
A linear sequence can be written in the form an = dn + c, where c is the zero-term value.
在KS3阶段，我们重点学习线性数列，即相邻项的差值是常数。这个常数称为公差，常用d表示。如果公差为正，数列递增；如果为负，数列递减。线性数列可以写成 an = dn + c 的形式，其中c是第零项的值。
3. The nth Term Formula | 第n项公式
ommon difference and b is the adjustment needed to match the first term. Here, an represents the term at position n.
第n项是一个公式，让你无需列出前面所有项就能计算出数列中的任意一项。对于线性数列，第n项的形式为 an = dn + b，其中d是公差，b是为了匹配第一项所需的调整数。an 表示位置n上的项。
4. How to Find the nth Term from a Pattern | 如何从模式中找到第n项
To find the nth term of a linear sequence, follow these steps:
- Identify the common difference (d) between consecutive terms.
- Write the sequence generated by dn (e.g. 3n, 4n, -2n).
- Compare this dn sequence with the original sequence to find the extra constant b.
- Write the nth term as an = dn + b.
要找到线性数列的第n项，请按照以下步骤：
- 找出相邻项之间的公差(d)。
- 写出由dn生成的数列（如3n, 4n, -2n）。
- 把这个dn数列与原数列进行比较，找出额外常数b。
- 将第n项写成 an = dn + b。
5. Worked Example 1: Simple Linear Sequence | 示例1：简单线性数列
Let's find the nth term of the sequence: 5, 8, 11, 14, 17, ...
Step 1: The common difference d = 8 − 5 = 3. So start with 3n.
Step 2: are with the original 5, 8, 11, 14, 17, ... Each term is exactly 2 more. Therefore, b = 2.
Step 3: The nth term is 3n + 2.
nth term: an = 3n + 2
让我们找出数列 5, 8, 11, 14, 17, ... 的第n项。
第1步：公差 d = 8 − 5 = 3。所以从 3n 开始。
第2步：3n 对应的数列是 3, 6, 9, 12, 15, ... 与原数列 5, 8, 11, 14, 17, ... 比较，每一项都恰好多了2。因此 b = 2。
第3步：第n项为 3n + 2。
第n项：an = 3n + 2
6. Worked Example 2: Sequence with Negative Common Difference | 示例2：公差为负数的数列
Consider the sequence: 10, 7, 4, 1, −2, ...
ng: 10 is 13 more than −3, 7 is 13 more than −6, etc. So b = 13. The nth term is −3n + 13.
an = −3n + 13
请考虑数列：10, 7, 4, 1, −2, ...
公差 d = 7 − 10 = −3。dn 数列为 −3n：−3, −6, −9, −12, −15, ... 比较：10 比 −3 多13，7 比 −6 多13，以此类推。所以 b = 13。第n项为 −3n + 13。
an = −3n + 13
7. Using the nth Term to Find Any Term | 利用第n项求任意项
ompute 3(50) + 2 = 150 + 2 = 152. This method works for any position, no matter how large.
一旦得到第n项公式，只需代入位置数字n即可。例如，对于第n项为 3n + 2 的数列，要求第50项，计算 3(50) + 2 = 150 + 2 = 152。无论n多大，这个方法都适用。
8. Generating Sequences from the nth Term | 由第n项生成数列
ins: 3, 8, 13, 18, 23, ... This is useful for checking whether a given number belongs to a sequence.
你也可以反过来操作：给出第n项，通过代入 n = 1, 2, 3, ... 生成前几项。例如，如果 an = 5n − 2，数列的开头几项为：3, 8, 13, 18, 23, ... 这对于检查某个给定的数是否属于该数列很有用。
9. Sequences with Fractions and Decimals | 含有分数和小数的数列
n = n + 0.5. For fractions like 1/2, 1, 3/2, 2, ... the difference is 1/2, giving an = ½n.
并非所有数列的项都是整数。有些可能涉及分数或小数，但求第n项的方法完全相同。例如：1.5, 2.5, 3.5, 4.5, ... 公差 d = 1，所以 an = n + 0.5。对于分数数列 1/2, 1, 3/2, 2, ... 公差为 1/2，得到 an = ½n。
10. Common Misconceptions | 常见误区
riginal to find b. Another error is forgetting to check that the nth term works for all positions—always test with n = 1, 2, and 3.
很多学生会把相邻项的差误当作 b 值。记住，b 不仅仅等于第一项。一定要将 dn 数列与原数列比较来求出 b。另一个错误是忘记检查第n项是否对所有位置都成立——务必用 n = 1, 2, 3 进行检验。
11. Practice Tips for KS3 Learners | KS3学习者的练习建议
When tackling exercises like those on Page 41, start by writing the sequence of differences. Practise with a variety of sequences: increasing, decreasing, and those starting from negative numbers. Draw a table linking position n (1, 2, 3...) to term values, which can help visualise the relationship. Use online interactive tools or flashcards to speed up mental arithmetic of the difference and b.
在做像第41页那样的练习题时，先写下相邻项的差。练习各种类型的数列：递增的、递减的，以及从负数开始的数列。画一个将位置 n (1, 2, 3...) 与各项数值联系起来，有助于直观理解关系。利用在线互动工具或记忆卡来加速公差和常数b的心算。
12. Summary | 总结
nerate new terms. Regular practice with exercises like those in your Cambridge workbook will make this skill automatic, preparing you for more advanced topics in algebra.
线性数列的第n项是一个强大的代数工具。通过找出公差 d 和零项调整数 b，你可以写出 an = dn + b，并用它求出任意项或生成新项。反复练习剑桥练习册中的类似题目，能使这一技能成为本能，为更高级的代数主题做好准备。
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