📚 IB Mathematics: Discrete Variables and Foundations of Sequences | IB数学：离散变量的概念与序列基础
In the IB Mathematics curriculum, understanding discrete variables is essential for modelling real-world situations where values change in distinct steps. Sequences, as ordered lists of numbers, provide a natural bridge between algebraic notation and discrete mathematical thinking.
在 IB 数学课程中，理解离散变量对于模拟那些以确定步长变化的现实情境至关重要。序列作为有序排列的数表，为代数记号与离散数学思维之间架起了一座天然的桥梁。
1. Discrete Variables vs. Continuous Variables | 离散变量与连续变量的对比
A discrete variable can only take specific, separated values. Examples include the number of students in a class, the score on a die, or the number of terms in a sequence. These values are countable and often integers, though they can also be other separated values.
离散变量只能取特定的、彼此分离的值。例如班级学生人数、骰子点数或序列中的项数。这些值是可数的，通常为整数，但也可能是其他分离的值。
A continuous variable can take any value within an interval, such as height, time, or temperature. When working with functions, a continuous domain is represented by a solid line, while a discrete domain is represented by distinct points.
连续变量可以取某个区间内的任意值，例如身高、时间或温度。在函数图像中，连续定义域用实线表示，而离散定义域则用一个个孤立的点表示。
- Discrete: number of terms n, index i, frequency counts.
- 离散：项数 n、下标 i、频数计数。
- Continuous: time t ∈ ℝ, weight, length.
- 连续：时间 t ∈ ℝ、重量、长度。
2. What Is a Sequence? | 什么是序列？
A sequence is an ordered set of numbers generated by a rule. Each number in a sequence is called a term. The first term is usually denoted by u₁ or a₁, the second by u₂, and so on.
序列是按特定规则生成的一组有序数字。序列中的每个数字称为项。第一项通常记为 u₁ 或 a₁，第二项记为 u₂，依此类推。
u₁, u₂, u₃, …, uₙ, …
The subscript n is a discrete variable: it takes positive integer values only. Thus a sequence is a function whose domain is the set of positive integers, usually ℤ⁺.
下标 n 是一个离散变量：它只取正整数。因此序列是定义在正整数集 ℤ⁺ 上的函数。
3. Sequence as a Discrete Function | 序列作为离散函数
If we write uₙ = f(n), the sequence can be viewed as a function f evaluated at n = 1, 2, 3, …. This distinguishes sequences from continuous functions like y = x², where x can be any real number.
若记 uₙ = f(n)，序列可看作函数 f 在 n = 1, 2, 3, … 处的函数值。这区分了序列与连续函数（如 y = x²，其中 x 可取任意实数）。
| Feature | Sequence | Continuous Function |
| Domain | ℤ⁺ (positive integers) | ℝ or an interval |
| Graph | Discrete points | Solid curve |
| Variable | Discrete n | Continuous x |
This idea is fundamental in IB: when graphing a sequence, you plot only the points with integer horizontal coordinates. Do not connect them unless the context explicitly allows interpolation.
这一思想在 IB 中至关重要：绘制序列图像时，只画横坐标为整数的点。除非上下文明确允许插值，否则不要将这些点连成线。
4. Notation and Terminology | 记号与术语
Common notation includes uₙ for the n-th term, d for common difference, r for common ratio, and Sₙ for the sum of the first n terms.
常见记号包括：uₙ 表示第 n 项，d 表示公差，r 表示公比，Sₙ 表示前 n 项和。
- a or u₁: first term | 首项
- d: common difference in arithmetic sequences | 等差数列的公差
- r: common ratio in geometric sequences | 等比数列的公比
- n: term number, a discrete variable | 项数，一个离散变量
In IB formula booklets, you might see uₙ = u₁ + (n − 1)d and uₙ = u₁r⁽ⁿ⁻¹⁾. These are the explicit formulas for arithmetic and geometric sequences.
在 IB 公式手册中，你会看到 uₙ = u₁ + (n − 1)d 和 uₙ = u₁r⁽ⁿ⁻¹⁾，它们分别是等差数列和等比数列的通项公式。
5. Arithmetic Sequences | 等差数列
An arithmetic sequence has a constant difference between consecutive terms. The difference, d, can be positive, negative, or zero.
等差数列的相邻两项之差恒定，该差值 d 可以是正数、负数或零。
uₙ = u₁ + (n − 1)d
For example, 3, 7, 11, 15, … has u₁ = 3 and d = 4. The 10th term is u₁₀ = 3 + 9 × 4 = 39.
例如 3, 7, 11, 15, … 中 u₁ = 3，d = 4。第 10 项为 u₁₀ = 3 + 9 × 4 = 39。
The graph of an arithmetic sequence lies on a straight line, but only the points with integer n are valid terms.
等差数列的图像位于同一条直线上，但只有整数 n 对应的点才是有效项。
6. Geometric Sequences | 等比数列
A geometric sequence has a constant ratio between consecutive terms. The common ratio r cannot be zero.
等比数列相邻两项之比恒定，公比 r 不能为零。
uₙ = u₁r⁽ⁿ⁻¹⁾
Example: 2, 6, 18, 54, … has u₁ = 2 and r = 3. Then u₅ = 2 × 3⁴ = 162.
例如：2, 6, 18, 54, … 中 u₁ = 2，r = 3，则 u₅ = 2 × 3⁴ = 162。
If r is negative, the terms alternate in sign. For instance, 5, −10, 20, −40, … has r = −2.
若 r 为负数，则各项符号交替。例如 5, −10, 20, −40, … 中 r = −2。
7. Recurrence Relations | 递推关系
A sequence can be defined recursively by giving the first term and a rule for later terms. For example, a recurrence relation for an arithmetic sequence is uₙ₊₁ = uₙ + d, with u₁ given.
序列也可以通过给出首项和后续项的规则来递归定义。例如，等差数列的一个递推关系是 uₙ₊₁ = uₙ + d，并给出 u₁。
uₙ₊₁ = ruₙ, with u₁ = a
This is the recurrence form for a geometric sequence. IB questions often ask you to find u₃ or u₄ from a recurrence relation, which reinforces the idea of a discrete step-by-step process.
这是等比数列的递推形式。IB 题目常要求从递推关系求 u₃ 或 u₄，以强化逐步离散过程的概念。
8. Sigma Notation and Series | 求和记号与级数
The sum of the first n terms of a sequence is called a series. Sigma notation provides a compact way to write such sums.
序列前 n 项之和称为级数。求和记号提供了一种紧凑的书写方式。
Σᵢ₌₁ⁿ uᵢ = u₁ + u₂ + … + uₙ
The index i is a discrete variable that runs over integer values from 1 to n. Sigma notation is frequently tested in IB papers, often combined with arithmetic or geometric series.
下标 i 是一个离散变量，从 1 取到 n。求和记号在 IB 试卷中经常出现，常与等差或等比级数结合。
- Arithmetic series: Sₙ = (n/2)[2a + (n − 1)d]
- 等差级数：Sₙ = (n/2)[2a + (n − 1)d]
- Geometric series: Sₙ = a(rⁿ − 1)/(r − 1), r ≠ 1
- 等比级数：Sₙ = a(rⁿ − 1)/(r − 1)，r ≠ 1
9. Applications in IB Problems | IB 题目中的应用
IB questions often require you to identify whether a situation is discrete and then apply sequence formulas. For example, compound interest with yearly compounding is geometric because each year the amount is multiplied by a constant factor 1 + i.
IB 题目常要求你判断情境是否为离散的，然后应用序列公式。例如按年复利计算的利息是等比数列，因为每年金额乘以恒定因子 1 + i。
Another example is depreciation: a car loses 12% of its value each year. Its value after n years is Vₙ = V₀(0.88)ⁿ. Since n is the number of years, it is a discrete variable.
另一个例子是折旧：汽车每年贬值 12%。其第 n 年的价值为 Vₙ = V₀(0.88)ⁿ。由于 n 是年数，它是一个离散变量。
In exam questions, you may be given a sequence such as uₙ = 5 × 2ⁿ and asked to find which term first exceeds 1000. This involves solving 5 × 2ⁿ > 1000, a typical discrete inequality.
考试中，你可能遇到给定序列 uₙ = 5 × 2ⁿ，并要求找出第一项超过 1000 的项数。这需要解不等式 5 × 2ⁿ > 1000，这是一个典型的离散不等式问题。
10. Common Mistakes and Pitfalls | 常见错误与陷阱
One common mistake is confusing the term number n with the term value uₙ. Another is applying the arithmetic formula to a geometric situation or vice versa.
常见错误之一是混淆项数 n 与项值 uₙ。另一个是对等比情境使用等差公式，或反之。
- Mistake: Using d instead of r for a geometric sequence.
- 错误：在等比数列中误用 d 而不是 r。
- Mistake: Starting at n = 0 instead of n = 1.
- 错误：从 n = 0 而不是 n = 1 开始。
- Mistake: Forgetting that Sₙ is the sum of n terms, not the n-th term.
- 错误：忘记 Sₙ 是前 n 项和而不是第 n 项。
Always check whether the sequence is arithmetic, geometric, or neither before choosing a formula. In IB, many questions purposely include a sequence that looks geometric but is not.
选择公式前，务必判断序列是等差、等比还是两者都不是。IB 中许多题目故意给出表面像等比但实际不是的序列。
11. Practice Questions | 练习
Let us test your understanding with a few short questions.
让我们用几个小问题来检验你的理解。
Question 1: A sequence is defined by uₙ = 3n + 2. Determine whether it is arithmetic or geometric, and find u₂₀.
问题 1：序列由 uₙ = 3n + 2 定义。判断它是等差数列还是等比数列，并求 u₂₀。
Solution: Since the difference uₙ₊₁ − uₙ = 3 is constant, it is arithmetic. u₂₀ = 3 × 20 + 2 = 62.
解：因为差值 uₙ₊₁ − uₙ = 3 恒定，所以是等差数列。u₂₀ = 3 × 20 + 2 = 62。
Question 2: Find the sum of the first 12 terms of the geometric sequence 3, 6, 12, …
问题 2：求等比数列 3, 6, 12, … 的前 12 项和。
Solution: a = 3, r = 2, n = 12. Thus S₁₂ = 3(2¹² − 1)/(2 − 1) = 3 × 4095 = 12285.
解：a = 3，r = 2，n = 12。因此 S₁₂ = 3(2¹² − 1)/(2 − 1) = 3 × 4095 = 12285。
Question 3: Is the sequence 1, 4, 9, 16, … arithmetic or geometric?
问题 3：序列 1, 4, 9, 16, … 是等差数列还是等比数列？
Solution: Neither. Differences are 3, 5, 7, … which are not constant, and ratios are 4, 2.25, 1.777… which are not constant.
解：两者都不是。差分为 3, 5, 7, … 不恒定，比值为 4, 2.25, 1.777… 也不恒定。
12. Conclusion | 结论
Discrete variables are the foundation of sequences in IB mathematics. Recognizing the differences between discrete and continuous settings helps you choose appropriate models, whether arithmetic, geometric, or recursive. Mastery of notation, formulas, and sigma notation will allow you to solve a wide range of IB problems with confidence.
离散变量是 IB 数学中序列的基础。识别离散情境与连续情境的差异有助于你选择合适的模型，无论是等差、等比还是递推模型。掌握记号、公式和求和记号，将让你自信地解决广泛的 IB 题目。
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