📚 Continuing Sequences | 延续序列
A sequence is a list of numbers arranged in a specific order according to a rule. In IGCSE mathematics, continuing a sequence means using the given terms to discover the pattern and then extending that pattern to find the next term or terms. Mastering this skill helps you recognise linear, quadratic, cubic, and special sequences, and it supports your understanding of the nth term formula.
序列是按特定规则排列的一组数。在 IGCSE 数学中，延续序列指的是利用已知的项找出规律，并将规律延伸以找出下一项或下几项。掌握这项技能能帮助你识别线性、二次、三次及特殊序列，也有助于你理解第 n 项公式。
1. What is a Sequence? | 什么是序列？
A sequence is an ordered set of numbers, often generated by a rule. Each number in the sequence is called a term. For example, in the sequence 2, 4, 6, 8, ..., the rule is "add 2 to the previous term". Knowing the rule allows you to continue the sequence and write the next terms: 10, 12, 14, and so on.
序列是一组有序的数，通常由某种规则生成。序列中的每一个数都叫做一项。例如，在序列 2, 4, 6, 8, … 中，规则是“前一项加 2”。知道了规则，你就能延续序列，写出下一项：10, 12, 14，以此类推。
In IGCSE exams, you will often be shown the first few terms of a sequence and asked to find the next term or the next two terms. To do this, you must first identify the type of sequence.
在 IGCSE 考试中，通常会给出序列的前几项，要求你找出下一项或下两项。要完成这一任务，你必须首先识别序列的类型。
2. How to Continue a Sequence | 如何延续序列
To continue a sequence, start by examining the differences between consecutive terms. If the difference is constant, the sequence is linear (arithmetic). If the first difference changes but the second difference is constant, the sequence is quadratic. For sequences defined by a term-to-term rule, simply apply the rule to the last known term.
要延续序列，首先要检查相邻两项之间的差值。如果差值是常数，那么该序列是线性（等差）序列。如果第一次差值在变化，但第二次差值是常数，那么该序列是二次序列。对于由递推规则定义的序列，只需将规则应用于最后已知项即可。
Use a systematic approach: write the terms in a row, then the first differences underneath. If those are not constant, find the second differences. Continue until you find a constant difference row. Then use that row to predict the next difference and ultimately the next term.
采用系统的方法：先在一行写下各项，再在其下写出第一次差分。如果这些差分不是常数，就继续求第二次差分，直到找到一组常数差分为止。然后利用这组常数差分来预测下一个差分，最终得到下一项。
3. Linear Sequences (Arithmetic) | 线性序列（等差序列）
... the common difference is 4. So the next term is 17 + 4 = 21. The following term is 25.
线性序列的一次差分为常数，这个常数被称为公差 d。要找出下一项，只需将 d 加到最后一项即可。例如，在序列 5, 9, 13, 17, … 中，公差为 4。因此下一项是 17 + 4 = 21，再下一项是 25。
The nth term of an arithmetic sequence is given by Tₙ = a + (n − 1)d te any term without listing all previous ones. In our example, Tₙ = 5 + 4(n − 1) = 4n + 1.
等差序列的第 n 项公式为 Tₙ = a + (n − 1)d，其中 a 是首项。如果你找到了第 n 项公式，也可以用公式直接生成任意项，而无需逐一列出之前的所有项。在我们的例子中，Tₙ = 5 + 4(n − 1) = 4n + 1。
When asked to continue a linear sequence, you can check the common difference in a quick table:
当要求延续线性序列时，你可以通过一个简表来检查公差：
| Term number (n) | 1 | 2 | 3 | 4 | 5 |
| Value | 5 | 9 | 13 | 17 | 21 |
| Diff. | +4 | +4 | +4 | +4 |
4. Quadratic Sequences | 二次序列
In a quadratic sequence, the first differences are not constant, but the second differences are constant. To continue the sequence, find the next first difference by adding the constant second difference, and then add that to the last term.
在二次序列中，第一次差分不是常数，但第二次差分是常数。要延续序列，就把常数第二次差分加到上一个第一次差分上，得到下一个第一次差分，再把该差分加到最后一项上。
Consider the sequence: 3, 6, 11, 18, 27, ... Let us set up a difference table:
考虑序列：3, 6, 11, 18, 27, … 我们建立差分表：
| n | 1 | 2 | 3 | 4 | 5 | 6 |
| Term | 3 | 6 | 11 | 18 | 27 | 38 |
| 1st diff | +3 | +5 | +7 | +9 | +11 | |
| 2nd diff | +2 | +2 | +2 | +2 |
gives 38. This method makes continuing quadratic sequences straightforward even without the nth term formula.
第二次差分恒为 2。下一个第一次差分将是 9 + 2 = 11。将此加到末项 27 上，得到 38。这种方法让延续二次序列变得简单直接，即便没有第 n 项公式也能完成。
any exam questions, you only need to continue a given quadratic pattern using differences.
如果你也了解通项公式 Tₙ = n² + 2，就能验证：T₆ = 6² + 2 = 38，结果吻合。但在许多考试题中，你只需要利用差分来延续给定的二次模式。
5. Cubic Sequences | 三次序列
Cubic sequences have constant third differences. To extend a cubic sequence, find the first, second, and third differences until the third differences become constant. Then work backwards to compute the next term.
三次序列具有常数第三次差分。要延伸一个三次序列，需先找出第一次、第二次和第三次差分，直到第三次差分成常数为止。然后再逆向计算得到下一项。
Take the cube numbers: 1, 8, 27, 64, 125, ... s are 7, 19, 37, 61. The second differences are 12, 18, 24. The third differences are 6, 6 — + 6 = 30, the next first difference is 61 + 30 = 91, and the next term is 125 + 91 = 216.
以立方数为例：1, 8, 27, 64, 125, … 第一次差分为 7, 19, 37, 61。第二次差分为 12, 18, 24。第三次差分为 6, 6 —— 是常数。因此，下一个第三次差分仍为 6，下一个第二次差分为 24 + 6 = 30，下一个第一次差分为 61 + 30 = 91，下一项为 125 + 91 = 216。
This technique works for any polynomial sequence. In IGCSE, cubic sequences may appear, but often you will be given enough terms to spot the constant difference in the third row. Draw the difference table carefully to avoid arithmetic mistakes.
这种方法适用于任何多项式序列。在 IGCSE 中，可能会遇到三次序列，但通常会给出足够多的项，让你能够在第三行发现常数差分。仔细绘制差分表，避免运算错误。
6. Fibonacci-type Sequences | 斐波那契型序列
continue, simply add the last two terms. So after 8 and 13, the next term is 8 + 13 = 21.
斐波那契型序列是指每一项都是前两项之和的序列。经典的斐波那契数列以 1, 1 开始，然后是 2, 3, 5, 8, 13, … 要延续该序列，只需将最后两项相加。因此，在 8 和 13 之后，下一项为 8 + 13 = 21。
You can also have sequences that are similar but start with different numbers. For instance, if the first two terms are 3 and 4, the sequence is 3, 4, 7, 11, 18, 29, ... The rule stays the same: add the last two terms. Examiners will often ask you to 'continue for two more terms', so make sure you use the newly found term to generate the next.
也可能遇到以不同数字开头的类似序列。例如，若前两项为 3 和 4，则序列为 3, 4, 7, 11, 18, 29, … 规则不变：将最后两项相加。考官经常会让你“再写出两项”，因此务必要用刚求出的项来生成下一项。
Look out for these sequences in word problems or diagrams: the growth of a plant, the number of steps, or a pattern of spirals can follow a Fibonacci-like rule.
在应用题或图示中留意此类序列：植物的生长、台阶数或螺旋图案都可能遵循类似斐波那契的规则。
7. Using Term-to-term Rules | 使用递推规则
continue such a sequence, you take the latest term, apply the rule, and get the next term.
有些序列不是由位置通项公式定义的，而是由递推规则定义的，例如写成 uₙ₊₁ = 2uₙ + 3 的形式。要延续这样的序列，你只需取出最后一项，应用规则，就能得到下一项。
Example: The first term is 5, and the rule is "multiply by 2 and subtract 1" ence begins 5, 9, 17, 33, ... To continue, compute 2 × 33 − 1 = 65, then 2 × 65 − 1 = 129. Always check whether the rule uses the immediately preceding term or earlier terms.
例子：首项为 5，规则是“乘以 2 再减 1”。也就是说 uₙ₊₁ = 2uₙ − 1。序列开头为 5, 9, 17, 33, … 要延续，计算 2 × 33 − 1 = 65，然后 2 × 65 − 1 = 129。务必检查规则使用的是紧前一项还是更靠前的项。
Sometimes the rule is given in words, and you must translate it into a mathematical operation. Take care with the order of operations and negative numbers.
有时规则以文字形式给出，你必须将其转化为数学运算。要特别注意运算顺序和负数的处理。
8. Special Sequences and Simple Patterns | 特殊序列与简单模式
There are common sequences you should recognise instantly, such as square numbers, triangular numbers, and powers of 2. Continuing these relies on knowing the underlying pattern.
- Square numbers: 1, 4, 9, 16, 25, ... (next is 36)
- Triangular numbers: 1, 3, 6, 10, 15, 21, ... (next is 28)
- Cube numbers: 1, 8, 27, 64, ... (next is 125)
- Powers of 2: 2, 4, 8, 16, 32, ... (next is 64)
还有一些你应该能立刻认出的常见序列，如平方数、三角数和 2 的幂次。延续这些序列依赖于理解其内在模式。
- 平方数：1, 4, 9, 16, 25, …（下一项是 36）
- 三角数：1, 3, 6, 10, 15, 21, …（下一项是 28）
- 立方数：1, 8, 27, 64, …（下一项是 125）
- 2 的幂次：2, 4, 8, 16, 32, …（下一项是 64）
Questions may also involve patterns of dots, matchsticks, or tiles. Count the number of objects in each step and then find the number sequence. For example, a pattern that grows by adding 3 each time gives a linear sequence, while one that grows by adding consecutive even numbers might be quadratic.
考题还可能涉及点阵、火柴棍或地砖等的模式。数出每一步中的物体数量，然后找出数字序列。例如，每次增加 3 个的模式形成线性序列，而每次增加连续偶数的模式则可能是二次序列。
9. Common Mistakes to Avoid | 常见错误
One common error is assuming a sequence is linear without checking the second difference. Always verify at least three differences before deciding the type. Another mistake is in arithmetic when finding differences; one small slip leads to the wrong next term.
一个常见错误是在没有检查第二次差分的情况下，就假定序列是线性的。在确定类型之前，至少要验证三个差分。另一个错误是求差分时的计算失误；一个小小的疏忽就会导致下一项错误。
Students also sometimes forget to use the newly found term in the next step of a Fibonacci-type sequence. For example, if asked for two more terms, you must use the first new term to find the second. Failing to do so gives only one correct extension.
学生们有时会忘记在斐波那契型序列的下一步中使用新求出的项。例如，如果要求写出接下来的两项，你必须用第一个新的项来求第二个新的项。若忘记这一点，就只能得到一个正确的延伸项。
With term-to-term rules, watch out for misapplying the order of operations, especially when multiplication and addition/subtraction are both present.
对于递推规则，要注意运算顺序，尤其是在乘法和加减法同时出现时。
10. Exam Tips for Continuing Sequences | 考试技巧：延续序列
Read the question carefully. It might ask for "the next term" or "the next two terms". Always show your working: draw the difference table and write down the constant difference you found. Even if your final answer is wrong, clear working earns method marks.
仔细读题。题目可能要求写出“下一项”或“下两项”。始终展示你的解题过程：画出差分表，并写下你发现的常数差分。即使最终答案有误，清晰的步骤也能为你赢得方法分。
When you finish, check your answer by verifying that the pattern holds for the new term. If you have an nth term formula, use it to double-check. But remember, continuing a sequence is a skill of its own — you do not always need a formula.
做完后，通过验证规律对新项是否成立来检查答案。如果你有第 n 项公式，可以用来复核。但请记住，延续序列本身就是一项技能——你并不总是需要公式。
Practice with a variety of sequences: linear, quadratic, cubic, Fibonacci-type, and term-to-term rules. The more you practise, the quicker you will spot the pattern and avoid common pitfalls.
练习各种类型的序列：线性序列、二次序列、三次序列、斐波那契型序列以及递推规则。练得越多，你就能越快发现模式，并避免常见陷阱。
Published by TutorHao | IGCSE Edexcel Maths Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图