Sequences and Series: A Comprehensive Guide for IB Mathematics | IB数学：数列与级数核心概念梳理

📚 Sequences and Series: A Comprehensive Guide for IB Mathematics | IB数学：数列与级数核心概念梳理

A sequence is an ordered list of numbers, while a series is the sum of the terms of a sequence. In IB Mathematics, sequences and series appear in both Analysis and Approaches and Applications and Interpretation, and they form the backbone of topics such as financial mathematics, growth and decay models, and binomial expansion. This guide organises the core concepts you need to master, from definitions to exam-ready strategies.

数列是按一定顺序排列的一列数，而级数则是将数列中的项相加所得的和。在IB数学中，数列与级数是分析与方法（AA）和应用与解释（AI）两个方向的共同重点，也是金融数学、增长与衰减模型、二项式展开等专题的基础。本指南系统梳理了需要掌握的核心概念，从定义到备考策略，帮助你高效复习。

1. What Are Sequences and Series? | 数列与级数的基本定义

A sequence is a function whose domain is the set of positive integers. Each value in the sequence is called a term, usually written as u₁, u₂, u₃, …, uₙ. The nth term or general term is an expression in terms of n that allows you to compute any term of the sequence directly.

数列是以正整数集为定义域的函数。数列中的每一个数值称为一项，通常记作 u₁、u₂、u₃、…、uₙ。第 n 项或通项是关于 n 的表达式，利用它可以直接计算数列中的任意一项。

A series is formed by adding the terms of a sequence. If you add the first n terms, you obtain the n-th partial sum, denoted Sₙ. A finite series has a finite number of terms, while an infinite series continues without end. Understanding the difference between a term and a sum is critical for applying the correct formulas.

级数是将数列中的各项相加得到的和。若将前 n 项相加，得到第 n 个部分和，记作 Sₙ。有限级数包含有限个项，而无穷级数则无限延续。分清“项”和“和”的概念是正确使用公式的前提。

uₙ = general term; Sₙ = u₁ + u₂ + … + uₙ

2. Arithmetic Sequences | 等差数列

An arithmetic sequence has a constant difference between consecutive terms. This constant difference is called the common difference, usually denoted d. If u₁ is the first term and d is the common difference, then the sequence is u₁, u₁ + d, u₁ + 2d, u₁ + 3d, …

等差数列相邻两项之差保持不变，这个常数称为公差，通常用 d 表示。若首项为 u₁，公差为 d，则数列为 u₁、u₁ + d、u₁ + 2d、u₁ + 3d、…。

The nth term of an arithmetic sequence is given by the formula below. This formula is essential for finding any specific term without listing all previous terms.

等差数列的通项公式如下。利用这个公式可以不必逐一列出前面各项，直接求出任意一项。

uₙ = u₁ + (n − 1)d

For example, the sequence 3, 7, 11, 15, … has u₁ = 3 and d = 4. Its 10th term is u₁₀ = 3 + 9 × 4 = 39. To determine whether a sequence is arithmetic, subtract consecutive terms and check whether the result is constant.

例如，数列 3、7、11、15、… 中 u₁ = 3，d = 4，第 10 项为 u₁₀ = 3 + 9 × 4 = 39。要判断一个数列是否为等差数列，只需计算相邻两项之差，看结果是否恒定。

3. Arithmetic Series | 等差级数

The sum of the first n terms of an arithmetic sequence is called an arithmetic series. There are two equivalent formulas for Sₙ. The first uses the first term u₁ and the common difference d; the second uses the first term u₁ and the nth term uₙ.

等差数列前 n 项的和称为等差级数。Sₙ 有两个等价公式：一个使用首项 u₁ 和公差 d；另一个使用首项 u₁ 和第 n 项 uₙ。

Sₙ = n/2 [2u₁ + (n − 1)d] = n/2 (u₁ + uₙ)

The second formula reminds us that the average of the first and last terms, multiplied by the number of terms, gives the sum. This is sometimes called the “pairing” method. Be careful: n represents the number of terms added, not the value of the last term index in a different context.

第二个公式的含义是：首项与末项的平均值乘以项数即为总和，这也被称为“配对”法。注意：n 表示相加的项数，不是末项本身的值。

If an arithmetic series has u₁ = 5, d = 3, and n = 20, then S₂₀ = 20/2 [2(5) + 19(3)] = 10(10 + 57) = 670. Always check your arithmetic when using these formulas, especially the factor (n − 1).

若等差级数中 u₁ = 5，d = 3，n = 20，则 S₂₀ = 20/2 [2(5) + 19(3)] = 10(10 + 57) = 670。使用公式时务必仔细检查运算，特别是 (n − 1) 这一因子。

4. Geometric Sequences | 等比数列

A geometric sequence has a constant ratio between consecutive terms. This ratio is called the common ratio, denoted r. If u₁ is the first term and r is the common ratio, the sequence is u₁, u₁r, u₁r², u₁r³, …

等比数列相邻两项之比保持不变，这个常数称为公比，通常用 r 表示。若首项为 u₁，公比为 r，则数列为 u₁、u₁r、u₁r²、u₁r³、…。

The nth term of a geometric sequence is found using the formula below. This formula is particularly important for modelling exponential growth and decay.

等比数列的通项公式如下。该公式在指数增长与衰减建模中特别重要。

uₙ = u₁ rⁿ⁻¹

For example, the sequence 2, 6, 18, 54, … has u₁ = 2 and r = 3. Its 6th term is u₆ = 2 × 3⁵ = 486. To find r, divide any term by the previous term: r = uₙ / uₙ₋₁.

例如，数列 2、6、18、54、… 中 u₁ = 2，r = 3，第 6 项为 u₆ = 2 × 3⁵ = 486。要确定公比，可用后一项除以前一项：r = uₙ / uₙ₋₁。

5. Geometric Series | 等比级数

The sum of the first n terms of a geometric sequence is called a geometric series. The formula below works for all finite geometric series as long as r ≠ 1. When r = 1, the series is simply n times u₁.

等比数列前 n 项的和称为等比级数。对于有限等比级数，只要 r ≠ 1，下面的公式均适用。当 r = 1 时，级数简化为 n × u₁。

Sₙ = u₁(1 − rⁿ) / (1 − r) = u₁(rⁿ − 1) / (r − 1)

Choose the form that keeps the denominator positive to reduce sign errors. For example, if u₁ = 3, r = 2, and n = 5, then S₅ = 3(2⁵ − 1)/(2 − 1) = 3(32 − 1) = 93.

选择使分母为正的形式可以减少符号错误。例如，若 u₁ = 3，r = 2，n = 5，则 S₅ = 3(2⁵ − 1)/(2 − 1) = 3(32 − 1) = 93。

When solving for n in a geometric series, you may need to use logarithms. For instance, if Sₙ = 93 and u₁ = 3, r = 2, then 2ⁿ = 32, so n = 5. Recognising powers of numbers can save time.

在等比级数中求解 n 时，通常需要运用对数。例如，若 Sₙ = 93，u₁ = 3，r = 2，则 2ⁿ = 32，因此 n = 5。识别常见数的幂可以节省时间。

6. Infinite Geometric Series | 无穷等比级数

An infinite geometric series has a finite sum only when the absolute value of the common ratio is less than 1, that is, |r| < 1. In this case, as n increases, rⁿ approaches 0, so the finite sum formula simplifies to the infinite sum formula below.

无穷等比级数只有在公比的绝对值小于 1，即 |r| < 1 时才有有限和。此时当 n 增大时，rⁿ 趋近于 0，因此有限和公式可以化简为下面的无穷和公式。

S∞ = u₁ / (1 − r), for |r| < 1

If |r| ≥ 1, the infinite series diverges, meaning it does not approach a finite limit. A common exam trap is applying the infinite sum formula without checking the condition. Always verify |r| < 1 before using S∞.

若 |r| ≥ 1，无穷级数发散，即它不趋近于任何有限值。考试中常见的陷阱是不检查条件就直接使用无穷和公式。使用 S∞ 之前务必验证 |r| < 1。

For example, the series 8 + 4 + 2 + 1 + … has u₁ = 8 and r = 1/2. Its infinite sum is S∞ = 8/(1 − 1/2) = 16. Fractions and decimals often appear in these questions, so keep values exact when possible.

例如，级数 8 + 4 + 2 + 1 + … 中 u₁ = 8，r = 1/2，其无穷和为 S∞ = 8/(1 − 1/2) = 16。此类问题常出现分数和小数，尽量保留精确值。

7. Sigma Notation | 求和符号

Sigma notation is a compact way to write series. The symbol ∑ means “sum”, and the notation ∑(i=1 to n) uᵢ represents the sum of uᵢ from i = 1 to i = n. The variable i is called the index, the lower limit is 1, and the upper limit is n.

求和符号是一种简洁的级数表示法。符号 ∑ 表示“求和”，记号 ∑(i=1 到 n) uᵢ 表示从 i = 1 到 i = n 对 uᵢ 求和。变量 i 称为指标，下限为 1，上限为 n。

Sigma notation follows two useful properties. First, the sum of a constant times a term equals the constant times the sum. Second, the sum of a sum or difference is the sum or difference of the individual sums.

求和符号有两个常用性质：其一，常数乘以项的和等于常数乘以项的和；其二，两项之和（或差）的和等于各自和的和（或差）。

∑(i=1 to n) c aᵢ = c ∑(i=1 to n) aᵢ; ∑(i=1 to n) (aᵢ ± bᵢ) = ∑(i=1 to n) aᵢ ± ∑(i=1 to n) bᵢ

When using sigma notation to evaluate arithmetic or geometric series, convert the general term into uᵢ form first. For example, ∑(i=1 to 10) (2i + 3) is an arithmetic series with u₁ = 5 and u₁₀ = 23, so its sum is 10/2 (5 + 23) = 140.

用求和符号计算等差或等比级数时，应先将通项写成 uᵢ 的形式。例如，∑(i=1 到 10) (2i + 3) 是等差数列，其中 u₁ = 5，u₁₀ = 23，因此总和为 10/2 (5 + 23) = 140。

8. Problem-Solving Strategies | 解题策略

Many exam problems ask you to find u₁, d, r, n, or Sₙ from given information. A reliable strategy is to write down the known formulas, identify the unknown values, and set up equations. For arithmetic sequences, use uₙ and Sₙ formulas; for geometric sequences, use uₙ and Sₙ formulas as well.

许多考试题目要求从已知条件中求出 u₁、d、r、n 或 Sₙ。可靠的策略是：先写出已知公式，找出未知量，再建立方程。等差数列使用 uₙ 与 Sₙ 公式，等比数列同样使用 uₙ 与 Sₙ 公式。

If you are given two terms and asked to find the sequence, write two equations in terms of u₁ and d (or u₁ and r), then solve them simultaneously. In geometric cases, dividing the equations often eliminates u₁ and gives r directly.

若已知两项并要确定整个数列，可以分别用 u₁ 和 d（或 u₁ 和 r）建立两个方程，然后联立求解。在等比数列中，将两个方程相除通常可以消去 u₁，直接求出 r。

Identify the type: check for constant difference or constant ratio.

判断类型：检查是否存在恒定公差或恒定公比。
List known quantities: u₁, d, r, n, uₙ, Sₙ, S∞.

列出已知量：u₁、d、r、n、uₙ、Sₙ、S∞。
Choose the relevant formula and solve step by step.

选择相关公式并逐步求解。
For geometric n, use logarithms after isolating the power.

对于等比数列的 n，先隔离幂，再用对数求解。

9. Applications in Finance and Science | 金融与科学中的应用

Arithmetic sequences and series model situations with constant change, such as straight-line depreciation or monthly fixed increases. Geometric sequences and series model percentage changes, compound interest, population growth, radioactive decay, and cooling temperatures.

等差数列和级数用于模拟恒定变化的情境，例如直线折旧或每月固定增长。等比数列和级数用于模拟百分比变化、复利、人口增长、放射性衰变和物体冷却等过程。

In IB Applications and Interpretation, compound interest is often written using the formula FV = PV(1 + i)ⁿ, where i is the interest rate per period. This is essentially a geometric sequence with first term PV and common ratio (1 + i).

在IB应用与解释课程中，复利公式常写作 FV = PV(1 + i)ⁿ，其中 i 是每期利率。这本质上是一个首项为 PV、公比为 (1 + i) 的等比数列。

Annuities and loan repayments can be modelled with geometric series: the total accumulated value is the sum of a geometric sequence. Make sure to align the time periods and interest rates, and round monetary values appropriately.

年金和贷款还款可以用等比级数建模：累积总额就是等比数列的和。务必确保时间周期与利率匹配，并按题目要求适当四舍五入货币数值。

10. Common Mistakes and Exam Tips | 常见错误与考试提示

One typical mistake is confusing the nth term with the sum of n terms. Read whether the question asks for uₙ or Sₙ. Another frequent error is using the infinite geometric sum formula when r is not between −1 and 1, producing a meaningless result.

一个典型错误是混淆第 n 项与前 n 项之和。做题时先确认题目要求的是 uₙ 还是 Sₙ。另一个常见错误是在 r 不介于 −1 和 1 之间时使用无穷等比和公式，导致结果无意义。

When using sigma notation, pay attention to the lower limit. If the index starts at 0 instead of 1, the number of terms is different, and the exponent in the geometric formula changes. Always check the bounds before applying formulas.

使用求和符号时，注意下限。如果指标从 0 开始而不是从 1 开始，则项数不同，等比公式中的指数也会改变。应用公式前务必检查求和上下限。

Finally, in IB exams, show your working clearly. Correct answers without method may not receive full marks, especially in questions assessing reasoning and problem-solving.

最后，在IB考试中，请清晰写出解题过程。仅有正确答案而没有方法展示，可能无法获得满分，尤其是考查推理和问题解决能力的题目。

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Sequences and Series: A Comprehensive Guide for IB Mathematics | IB数学：数列与级数核心概念梳理