Sequences and Series: Arithmetic, Geometric and Sigma Notation | 数列与级数：等差、等比与求和符号

📚 Sequences and Series: Arithmetic, Geometric and Sigma Notation | 数列与级数：等差、等比与求和符号

In Edexcel A Level Mathematics, sequences and series appear across pure mathematics, from basic nth term rules to sums of geometric progressions and sigma notation. This article covers the key definitions, formulas, and problem types you need for the exam.

在 Edexcel A Level 数学中，数列与级数贯穿纯数学的多个主题，从基本第 n 项公式到等比数列求和以及求和符号。本文涵盖考试所需的核心定义、公式和常见题型。

1. Key Definitions and Notation | 核心定义与符号

A sequence is an ordered list of numbers u₁, u₂, u₃, … following a rule. A series is the sum of the terms of a sequence. The nth term is usually written uₙ or aₙ.

数列是按规则排列的一列数 u₁、u₂、u₃……；级数是数列各项之和。第 n 项通常写作 uₙ 或 aₙ。

You must be comfortable with subscript notation: u₁ is the first term, u₂ is the second term, and Sₙ means the sum of the first n terms: Sₙ = u₁ + u₂ + … + uₙ.

你必须熟悉下标符号：u₁ 是第一项，u₂ 是第二项，Sₙ 表示前 n 项之和：Sₙ = u₁ + u₂ + … + uₙ。

• Sequence: ordered list; Series: sum of terms | 数列：有序列表；级数：各项之和
• uₙ: nth term; Sₙ: sum of first n terms | uₙ：第 n 项；Sₙ：前 n 项和

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2. Arithmetic Sequences: nth Term | 等差数列：第 n 项

An arithmetic sequence adds a constant difference d each time. If the first term is a, then the nth term is given by:

等差数列每次加上固定公差 d。若首项为 a，则第 n 项为：

uₙ = a + (n − 1)d

Here d = uₙ₊₁ − uₙ. You can use this formula to find any term, or to find a and d from given terms.

其中 d = uₙ₊₁ − uₙ。你可以用该公式求任意项，或由已知项反求 a 和 d。

Example: In the sequence 3, 7, 11, 15, …, the first term is 3 and d = 4. The 10th term is u₁₀ = 3 + (10 − 1) × 4 = 39.

例如：数列 3、7、11、15……中，首项 a = 3，d = 4。第 10 项为 u₁₀ = 3 + (10 − 1) × 4 = 39。

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3. Arithmetic Series: Sum Formula | 等差级数：求和公式

The sum of the first n terms of an arithmetic series can be found using either form:

等差级数前 n 项和可用以下任一形式：

Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l)

where l = a + (n − 1)d is the last term. The second form is useful when you know the first and last terms.

其中 l = a + (n − 1)d 为末项。第二种形式在你已知首项和末项时很方便。

Example: Find the sum of the first 20 terms of 5 + 8 + 11 + … . Here a = 5, d = 3, so S₂₀ = 20/2 [2(5) + 19(3)] = 10 × 67 = 670.

例如：求 5 + 8 + 11 + … 前 20 项之和。这里 a = 5，d = 3，所以 S₂₀ = 20/2 [2(5) + 19(3)] = 10 × 67 = 670。

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4. Geometric Sequences: nth Term | 等比数列：第 n 项

A geometric sequence multiplies by a constant ratio r each time. If the first term is a, the nth term is:

等比数列每次乘以固定公比 r。若首项为 a，则第 n 项为：

uₙ = a rⁿ⁻¹

where r = uₙ₊₁ / uₙ. Be careful: the power is n − 1, not n.

其中 r = uₙ₊₁ / uₙ。注意：指数是 n − 1，不是 n。

Example: In 2, 6, 18, 54, …, a = 2 and r = 3. The 8th term is u₈ = 2 × 3⁷ = 4374.

例如：在 2、6、18、54……中，a = 2，r = 3。第 8 项为 u₈ = 2 × 3⁷ = 4374。

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5. Geometric Series: Finite Sum | 等比级数：有限和

The sum of the first n terms of a geometric series is:

等比级数前 n 项和为：

Sₙ = a(1 − rⁿ) / (1 − r)

for r ≠ 1. This formula can also be written as Sₙ = a(rⁿ − 1)/(r − 1). Use the version that avoids negative denominators.

其中 r ≠ 1。该公式也可写作 Sₙ = a(rⁿ − 1)/(r − 1)。选择能避免负分母的形式。

Example: Find the sum of the first 6 terms of 3 + 6 + 12 + … . Here a = 3, r = 2, so S₆ = 3(1 − 2⁶)/(1 − 2) = 3(1 − 64)/(−1) = 189.

例如：求 3 + 6 + 12 + … 前 6 项之和。这里 a = 3，r = 2，所以 S₆ = 3(1 − 2⁶)/(1 − 2) = 3(1 − 64)/(−1) = 189。

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6. Infinite Geometric Series and Sum to Infinity | 无穷等比级数与无穷和

If |r| < 1, the terms get smaller and the infinite geometric series converges to a finite sum:

若 |r| < 1，项会越来越小，无穷等比级数收敛到有限和：

S∞ = a / (1 − r)

This only works when −1 < r < 1. If |r| ≥ 1, the sum diverges and has no finite value.

该公式仅在 −1 < r < 1 时成立。若 |r| ≥ 1，级数发散，不存在有限值。

Example: 10 + 5 + 2.5 + … has a = 10, r = 1/2, so S∞ = 10/(1 − 1/2) = 20.

例如：10 + 5 + 2.5 + … 中 a = 10，r = 1/2，所以 S∞ = 10/(1 − 1/2) = 20。

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7. Sigma Notation and Properties | 求和符号及其性质

Sigma notation compactly represents a sum. For example, ∑ from k = 1 to 5 of (2k + 1) means 3 + 5 + 7 + 9 + 11 = 35.

求和符号可简洁表示和。例如，∑ 从 k = 1 到 5 的 (2k + 1) 表示 3 + 5 + 7 + 9 + 11 = 35。

∑ₖ₌₁⁵ (2k + 1) = 3 + 5 + 7 + 9 + 11 = 35

You can use standard sum formulas to evaluate arithmetic and geometric series written in sigma notation. Always check the starting value of k and the number of terms.

你可以利用标准求和公式计算用求和符号表示的等差或等比级数。务必检查 k 的起始值和项数。

Properties: ∑(c aₖ) = c ∑aₖ and ∑(aₖ + bₖ) = ∑aₖ + ∑bₖ for a constant c.

性质：∑(c aₖ) = c ∑aₖ，并且 ∑(aₖ + bₖ) = ∑aₖ + ∑bₖ，其中 c 为常数。

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8. Recurrence Relations | 递推关系

A recurrence relation defines each term from the previous one, such as uₙ₊₁ = 2uₙ − 1, with u₁ = 3. This can generate the sequence 3, 5, 9, 17, …

递推关系通过前一项定义后一项，例如 uₙ₊₁ = 2uₙ − 1，且 u₁ = 3。这可以生成数列 3、5、9、17……

Be comfortable with both increasing and decreasing sequences, and with finding a term after several iterations.

要熟悉递增和递减数列，并能多次迭代求出后续项。

Sometimes you may need to show that a sequence is periodic or tends to a limit. Check successive terms carefully.

有时你可能需要证明数列具有周期性或趋于某个极限。仔细检查连续项。

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9. Applications and Modelling | 应用与建模

Sequences and series model savings, population growth, radioactive decay, and geometric patterns. Arithmetic models fit constant additive change; geometric models fit constant percentage change.

数列与级数可用于建模储蓄、人口增长、放射性衰变和几何图案。等差模型适用于固定增加量；等比模型适用于固定百分比变化。

For example, if a savings account pays 5% interest per year, the balance after each year forms a geometric sequence with r = 1.05. The sum of deposits can be found using Sₙ.

例如，如果储蓄账户每年支付 5% 利息，每年余额构成公比 r = 1.05 的等比数列。存款总和可用 Sₙ 计算。

Always define your variables and state whether r or d is positive or negative in context.

始终定义变量，并根据题意说明 r 或 d 的正负。

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10. Exam Tips and Common Mistakes | 考试技巧与常见错误

Common mistakes include using n instead of n − 1 in nth term formulas, applying S∞ when |r| ≥ 1, and miscounting the number of terms in sigma notation.

常见错误包括：第 n 项公式中误用 n 而不是 n − 1；在 |r| ≥ 1 时使用 S∞；以及在求和符号中数错项数。

• Check if the sequence is arithmetic (add/subtract) or geometric (multiply/divide).
• 确认数列是等差（加减）还是等比（乘除）。
• Write down a, d or r, and n before substituting.
• 代入前先写出 a、d 或 r、n。
• For sum to infinity, verify |r| < 1.
• 使用无穷和公式前，验证 |r| < 1。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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