📚 Continuing Sequences: How to Write the Next Terms from a Pattern | 序列的延续：如何根据规律写出后续项
In IGCSE Mathematics, one of the most fundamental skills is recognising a pattern in a sequence and using that pattern to find the next terms. A sequence is an ordered list of numbers, and each number in the list is called a term. To continue a sequence correctly, you must identify the rule that connects one term to the next, or the rule that connects each term to its position in the sequence.
在IGCSE数学中，最基本的技能之一就是识别序列中的规律，并利用该规律写出后续项。序列是按顺序排列的一组数，其中的每一个数称为一项。要正确地延续一个序列，你必须找出连接相邻两项的规则，或者找出每一项与其所在位置之间的规则。
1. What Is a Sequence? | 什么是序列？
A sequence is a set of numbers written in a definite order. For example, 3, 6, 9, 12, ... is a sequence where each term increases by 3. The three dots "..." mean that the sequence continues indefinitely. Each number in the sequence is called a "term": the first term is often written as a1, the second term as a2, and so on.
序列是按确定顺序写出的一组数。例如，3, 6, 9, 12, ... 就是一个每一项都增加3的序列。三个点"..."表示该序列无限延续下去。序列中的每一个数称为一项：第一项通常记作 a1，第二项记作 a2，以此类推。
In your Edexcel IGCSE exam, you may be asked to find the next two or three terms, to describe the rule in words, or to write down the nth term formula. All of these tasks start with the same first step: observe the pattern carefully.
在Edexcel IGCSE考试中，你可能会被要求写出接下来的两项或三项、用文字描述规律，或者写出第n项公式。所有这些任务都以同一个第一步开始：仔细观察规律。
2. Common Types of Sequences | 常见的序列类型
Before you can continue a sequence, you need to recognise its type. The most common types at IGCSE level are: linear (arithmetic) sequences, quadratic sequences, geometric sequences, and special sequences such as square numbers, cube numbers, and Fibonacci-like sequences.
在延续一个序列之前，你需要识别它的类型。IGCSE阶段最常见的类型有：线性（等差）序列、二次序列、等比序列，以及特殊序列，如平方数、立方数和类斐波那契序列。
- Linear sequence: the difference between consecutive terms is constant. Example: 5, 8, 11, 14, ...
- Linear sequence (等差): 相邻两项的差恒定。例如：5, 8, 11, 14, ...
- Quadratic sequence: the second difference (difference between the differences) is constant. Example: 1, 4, 9, 16, ...
- Quadratic sequence (二次序列): 二阶差（差值的差）恒定。例如：1, 4, 9, 16, ...
- Geometric sequence: each term is multiplied by a fixed number. Example: 2, 6, 18, 54, ...
- Geometric sequence (等比序列): 每一项乘以一个固定的数。例如：2, 6, 18, 54, ...
- Special pattern: for example, Fibonacci sequence where each term is the sum of the two previous terms.
- 特殊规律: 例如斐波那契序列，其中每一项是前两项之和。
3. Continuing a Linear Sequence | 延续等差序列
In a linear sequence, you find the common difference by subtracting one term from the next. Once you have this difference, you simply add it to the last known term to find the next term, and repeat as many times as required.
在等差序列中，通过用后一项减去前一项来求出公差。一旦求出公差，只需把它加到最后一个已知项上就能得到下一项，并按需要重复操作。
Example: 7, 12, 17, 22, ...
例如：7, 12, 17, 22, ...
erence is 12 − 7 = 5. So the next term is 22 + 5 = 27, then 27 + 5 = 32, then 32 + 5 = 37.
差值为12 − 7 = 5。所以下一项是22 + 5 = 27，然后是27 + 5 = 32，再然后是32 + 5 = 37。
Always check that the difference is the same between every pair of consecutive terms. If the difference changes, the sequence is not linear.
一定要检查每一对相邻项之间的差是否相同。如果差值发生变化，该序列就不是等差序列。
4. Using the First Difference Table | 使用一阶差表
When a sequence is more complicated, writing a difference table helps you see the pattern clearly. Write the terms in a row, then underneath write the differences between consecutive terms.
当序列更复杂时，写出差表可以帮助你清楚地看到规律。把各项写在一行，在每一项下方写出相邻两项的差。
| Term number | 1 | 2 | 3 | 4 | 5 |
| Term | 3 | 6 | 11 | 18 | ? |
| First difference | — | 3 | 5 | 7 | ? |
2 each time. So the next difference is 7 + 2 = 9. Therefore the next term is 18 + 9 = 27.
差值为3, 5, 7，每次增加2。所以下一个差值是7 + 2 = 9。因此下一项是18 + 9 = 27。
This method is especially useful for quadratic sequences, where the first differences form a linear sequence.
这种方法在二次序列中尤其有用，因为此时一阶差构成等差序列。
5. Quadratic Sequences and Second Differences | 二次序列与二阶差
A quadratic sequence has a constant second difference. This means the first differences do not stay the same, but the differences between the first differences do stay constant.
二次序列具有恒定的二阶差。这意味着其一阶差并不相同，但一阶差之间的差值保持恒定。
Example: 4, 7, 12, 19, 28, ...
例如：4, 7, 12, 19, 28, ...
First differences: 3, 5, 7, 9 (increasing by 2). Second difference: 2 (constant).
一阶差：3, 5, 7, 9（每次增加2）。二阶差：2（恒定）。
st differences: the next first difference is 9 + 2 = 11, so the next term is 28 + 11 = 39.
要求下一项，继续一阶差：下一个一阶差为9 + 2 = 11，所以下一项为28 + 11 = 39。
First difference: an+1 - an | Second difference: (an+2 - an+1) - (an+1 - an)
In the Edexcel IGCSE exam, you are often asked to find the next two terms of a quadratic sequence. You must show the first differences and then the second difference to justify your answer.
在Edexcel IGCSE考试中，你经常被要求写出二次序列接下来的两项。你必须写出一阶差和二阶差来证明你的答案。
6. Geometric Sequences and the Common Ratio | 等比序列与公比
A geometric sequence is one where each term is multiplied by the same number to get the next term. This number is called the common ratio. To find the common ratio, divide any term by the previous term.
等比序列是指每一项都乘以同一个数得到下一项的序列。这个数称为公比。要求公比，用任意一项除以前一项即可。
Example: 5, 15, 45, 135, ...
例如：5, 15, 45, 135, ...
s 15 ÷ 5 = 3. So the next term is 135 × 3 = 405, and the following term is 405 × 3 = 1215.
公比为15 ÷ 5 = 3。所以下一项是135 × 3 = 405，再下一项是405 × 3 = 1215。
Be careful with negative ratios. If the ratio is negative, the terms alternate between positive and negative. For example: 2, −6, 18, −54, ... has common ratio −3.
注意负公比的情况。如果公比为负数，各项会在正负之间交替。例如：2, −6, 18, −54, ... 的公比为−3。
Common ratio r = an+1⁄an
7. Special Patterns: Square, Cube, and Pascal Sequences | 特殊规律：平方、立方与帕斯卡序列
Some sequences are based on familiar mathematical forms. Recognising these can save you time in an exam.
有些序列基于常见的数学形式。识别这些形式可以在考试中节省时间。
- Square numbers: 1, 4, 9, 16, 25, ... The pattern is 12, 22, 32, 42, .... The next terms are 36, 49.
- 平方数： 1, 4, 9, 16, 25, ... 规律是12, 22, 32, 42, ...。后续项为36、49。
- Cube numbers: 1, 8, 27, 64, ... The pattern is 13, 23, 33, 43, .... The next terms are 125, 216.
- 立方数： 1, 8, 27, 64, ... 规律是13, 23, 33, 43, ...。后续项为125、216。
- Fibonacci sequence: 1, 1, 2, 3, 5, 8, ... Each term is the sum of the two previous terms. The next terms are 13, 21.
- 斐波那契序列： 1, 1, 2, 3, 5, 8, ... 每一项是前两项之和。后续项为13、21。
- Triangular numbers: 1, 3, 6, 10, 15, ... The pattern is 1, 1+2, 1+2+3, 1+2+3+4, ... The next terms are 21, 28.
- 三角数： 1, 3, 6, 10, 15, ... 规律为1, 1+2, 1+2+3, 1+2+3+4, ... 后续项为21、28。
These patterns often appear as part of a larger question, so learning them by heart is very helpful.
这些规律经常出现在大题中，因此熟记它们非常有帮助。
8. Using the Position-to-Term Rule (nth Term) | 使用“位置到项”规则（第n项）
To continue a sequence without listing terms one by one, you can use the nth term formula. For a linear sequence, the nth term is of the form an + b, where a is the common difference and b is a constant.
为了不逐项列出序列，你可以使用第n项公式。对于等差序列，第n项的形式为an + b，其中a是公差，b是常数。
Example: For the sequence 5, 8, 11, 14, ... the common difference is 3. The nth term is 3n + b. Since the first term is 5, we set 3(1) + b = 5, so b = 2. The nth term is 3n + 2.
例如：对于序列5, 8, 11, 14, ...，公差为3。第n项为3n + b。由于第一项是5，令3(1) + b = 5，所以b = 2。第n项为3n + 2。
To find the next term after the ones you have been given, substitute the required position number into the formula.
在已知项之后继续求下一项时，只需把所需的位置编号代入公式即可。
9. Finding the nth Term of a Quadratic Sequence | 求二次序列的第n项
For a quadratic sequence where the second difference is 2d, the nth term is dn2 + en + f. To find d, take half of the second difference. Then use the known first term and second term to solve for e and f.
对于二阶差为2d的二次序列，其第n项为dn2 + en + f。要求d，取二阶差的一半。然后利用已知的第一项和第二项解出e和f。
Example: Sequence 4, 7, 12, 19, ... Second difference = 2, so d = 1. The nth term is n2 + en + f.
例如：序列4, 7, 12, 19, ... 二阶差 = 2，所以d = 1。第n项为n2 + en + f。
When n = 1: 1 + e + f = 4, so e + f = 3. When n = 2: 4 + 2e + f = 7, so 2e + f = 3. Subtracting gives e = 0. Then f = 3. The nth term is n2 + 3.
当n = 1时：1 + e + f = 4，所以e + f = 3。当n = 2时：4 + 2e + f = 7，所以2e + f = 3。相减得e = 0。于是f = 3。第n项为n2 + 3。
an = n2 + 3
You can check this: 12 + 3 = 4, 22 + 3 = 7, 32 + 3 = 12, 42 + 3 = 19. Correct.
你可以检验：12 + 3 = 4，22 + 3 = 7，32 + 3 = 12，42 + 3 = 19。正确。
10. Worked Example: Mixed Pattern with Alternating Differences | 综合示例：差值交替变化的规律
Consider the sequence: 2, 3, 5, 8, 12, ... The differences are 1, 2, 3, 4, which increase by 1 each time. This is actually a quadratic-type pattern because the differences form a linear sequence.
考虑序列：2, 3, 5, 8, 12, ... 其一阶差为1, 2, 3, 4，每次增加1。这实际上是二次型规律，因为一阶差构成等差序列。
term is 12 + 5 = 17. The following difference is 6, so the term after that is 17 + 6 = 23.
下一个差值为5，所以下一项为12 + 5 = 17。再下一个差值为6，所以再下一项为17 + 6 = 23。
Alternatively, you can find the nth term using the quadratic method. Second difference = 1, so d = 0.5. The nth term is 0.5n2 + en + f. Solving gives an = 0.5n2 + 0.5n + 1. Check: for n = 1, 0.5 + 0.5 + 1 = 2. Correct.
或者，你可以用二次方法求第n项。二阶差 = 1，所以d = 0.5。第n项为0.5n2 + en + f。解得an = 0.5n2 + 0.5n + 1。检验：当n = 1时，0.5 + 0.5 + 1 = 2。正确。
11. Common Mistakes and How to Avoid Them | 常见错误及避免方法
Students often make simple errors when continuing sequences. Here are the most common ones and the strategies to avoid them.
学生在延续序列时经常会犯一些简单错误。以下是最常见的错误以及避免策略。
- Mistake 1: Assuming the difference stays the same when it does not.
- 错误1： 假定差值不变，但实际上差值在变化。
- Mistake 2: Forgetting to check the second difference for quadratic sequences.
- 错误2： 对于二次序列忘记检查二阶差。
- Mistake 3: Misreading a geometric sequence as linear because the numbers grow quickly.
- 错误3： 因为数字增长很快，误把等比序列当作等差序列。
- Mistake 4: Forgetting the rule for Fibonacci-type sequences, especially when the first term is not 1.
- 错误4： 忘记斐波那契型序列的规则，尤其当第一项不是1时。
- Mistake 5: Not showing working in the exam. Always show the differences clearly.
- 错误5： 考试中不写过程。一定要清楚地写出差值。
The best way to avoid these mistakes is to always ask yourself: "Is the difference constant? Is the ratio constant? Or is the second difference constant?"
避免这些错误最好的方法是始终问自己：“差值恒定吗？比值恒定吗？还是二阶差恒定？”
12. Summary and Exam Tips | 总结与考试技巧
To continue any sequence successfully, follow these steps. First, look at the differences or ratios between consecutive terms. Second, identify the type of sequence: linear, quadratic, geometric, or special. Third, use the rule to generate the required next terms. Finally, check your answer by making sure the pattern continues consistently.
要成功延续任何序列，请遵循以下步骤。首先，观察相邻两项之间的差或比值。其次，识别序列类型：等差、二次、等比或特殊序列。第三，利用该规律生成所需的后续项。最后，通过检查规律是否一致来验证答案。
In the Edexcel IGCSE exam, pay attention to the number of terms you are asked to write. If the question says "next two terms", do not write three. Also remember to show your working, especially difference tables, because method marks are important even if your final answer is slightly wrong.
在Edexcel IGCSE考试中，注意题目要求你写几项。如果题目要求“接下来的两项”，不要写三项。另外要记住写出过程，特别是差表，因为即使最终答案略有错误，方法分也很重要。
Finally, practise with a wide variety of sequences. The more patterns you have seen, the faster you will recognise them in the exam.
最后，多练习各种类型的序列。你见过的规律越多，在考试中识别它们的速度就越快。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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