📚 Geometric Sequences | 几何数列
Geometric sequences are one of the fundamental building blocks in A-Level Mathematics. Understanding how they grow, how to sum their terms, and how they apply to real-world scenarios like compound interest is essential for success in the Edexcel syllabus. This article will guide you through every key concept, from the nth term formula to the sum to infinity.
几何数列是A-Level数学中的基础模块之一。理解它们如何增长、如何求和以及如何应用于复利等现实场景，对于在Edexcel考试大纲中取得成功至关重要。本文将带你逐一掌握从通项公式到无穷等比级数求和的所有核心概念。
1. Definition of Geometric Sequences | 几何数列的定义
A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio, denoted by r. For example, in the sequence 2, 6, 18, 54, ... the common ratio is 3.
几何数列是指从第二项起，每一项都是前一项乘以一个固定的非零常数（称为公比，记作 r）得到的数列。例如，数列 2, 6, 18, 54, ... 的公比就是 3。
If the first term is denoted as a, the terms of a geometric sequence can be written as: a, ar, ar², ar³, ..., arⁿ⁻¹, where n is the position of the term.
如果首项记作 a，那么几何数列的各项可以写成：a, ar, ar², ar³, ..., arⁿ⁻¹，其中 n 表示项的位置。
2. The nth Term Formula | 通项公式
re a is the first term, r is the common ratio, and n is the term number. This is provided in the Edexcel formula booklet but you must know how to use it correctly.
几何数列的第 n 项公式为：uₙ = arⁿ⁻¹，其中 a 是首项，r 是公比，n 是项数。Edexcel 的公式手册会提供此公式，但你仍需知道如何正确使用它。
nce starts at n = 0, the formula would be arⁿ, but typically in A-Level we start at n = 1.
注意，n 是从 1 开始的正整数。如果数列从 n = 0 开始，公式会变成 arⁿ，但 A-Level 中通常从 n = 1 开始。
3. Examples of Finding the nth Term | 求通项公式举例
metric sequence 5, 15, 45, ... . First, identify a = 5 and r = 15 ÷ 5 = 3. Then u₆ = 5 × 36-1 = 5 × 3⁵ = 5 × 243 = 1215.
例1：求几何数列 5, 15, 45, ... 的第 6 项。首先，确定 a = 5，r = 15 ÷ 5 = 3。那么 u₆ = 5 × 36-1 = 5 × 3⁵ = 5 × 243 = 1215。
Dividing the equations: (ar⁴)/(ar²) = 4/16 → r² = 1/4, so r = ±1/2. If r = 1/2, then a × (1/2)² = 16 → a/4 = 16 → a = 64. If r = -1/2, then a × (1/4) = 16 → a = 64 as well. So there are two possible sequences.
例2：某几何数列的第三项为 16，第五项为 4。求首项和公比。利用 u₃ = ar² = 16 和 u₅ = ar⁴ = 4。两式相除：(ar⁴)/(ar²) = 4/16 → r² = 1/4，所以 r = ±1/2。若 r = 1/2，则 a × (1/2)² = 16 → a/4 = 16 → a = 64。若 r = -1/2，同样 a = 64。因此存在两个可能的数列。
4. Sum of the First n Terms | 前 n 项和
ively, Sₙ = a(rⁿ - 1)/(r - 1), which is often more convenient when r > 1. Both are listed in the formula booklet.
几何级数（部分和数列）的前 n 项和记作 Sₙ。公式为：Sₙ = a(1 - rⁿ)/(1 - r)，当 r ≠ 1 时。或者 Sₙ = a(rⁿ - 1)/(r - 1)，当 r > 1 时通常更方便。公式手册中两种形式都有。
constant, and Sₙ = na. This is a trivial case but worth noting to avoid division by zero.
如果 r = 1，数列为常数列，则 Sₙ = na。这种情况比较简单，但值得注意以避免除以零。
5. Sum to Infinity of a Convergent Series | 收敛级数的无穷和
When the absolute value of the common ratio is less than 1, i.e., |r| rmula is S∞ = a / (1 - r). This is derived from the Sₙ formula as n → ∞, since rⁿ → 0 when |r| < 1.
当公比的绝对值小于 1，即 |r| < 1 时，几何级数收敛，我们可以求出它的无穷和 S∞。公式为 S∞ = a / (1 - r)。这是由 Sₙ 公式在 n → ∞ 时导出的，因为当 |r| < 1 时 rⁿ → 0。
A common exam trick is to give a series that looks like it might diverge but has |r| of the series 12 + 6 + 3 + ... to infinity has a = 12, r = 1/2, so S∞ = 12/(1 - 1/2) = 24.
考试中常见的陷阱是给出一个看似发散但 |r| 略小于 1 的级数，或者要求从非首项开始求和。例如，级数 12 + 6 + 3 + ... 的无穷和：a = 12, r = 1/2，则 S∞ = 12/(1 - 1/2) = 24。
6. Geometric Series Problems | 几何级数应用题
Geometric sequences often appear in financial contexts such as compound interest and depreciation, as well as in modeling population growth or radioactive decay. The key is to identify the first term and the common ratio from the context.
几何数列经常出现在财务背景中，如复利和折旧，以及模拟人口增长或放射性衰变。关键是从情境中识别出首项和公比。
For instance, an investment of £1000 grows at a fixed rate of 5% year. The amount after n years forms a geometric sequence with a = 1000(1.05) after year 1? Wait, careful: if the initial amount is £ he nth term, uₙ = 1000 × 1.05ⁿ. Some textbooks treat the initial amount as term 0, then uₙ = 1000 × 1.05ⁿ for n years. Both interpretations are valid as long as you define n clearly. The sum of the amounts over n years is not typically asked, but the sum of geometric series can represent total savings with regular deposits.
例如，一项 1000 英镑的投资每年固定增长 5%。历年金额构成几何数列：第 1 年后为 1000 × 1.05 = 1050，所以 a = 1050，r = 1.05。n 年后的金额为第 n 项 uₙ = 1000 × 1.05ⁿ。有些教材把初始金额作为第 0 项，则 n 年后的金额为 uₙ = 1000 × 1.05ⁿ。只要明确定义 n，两种解释都可以。历年金额的总和通常不考，但几何级数求和可以表示定期存款的总额。
7. Proof of Sum Formula | 求和公式的证明
iply both sides by r: rSₙ = ar + ar² + ar³ + ... + arⁿ. Subtract the second equation from the first: Sₙ - rSₙ = a - arⁿ → Sₙ(1 - r) = a(1 - rⁿ). Therefore, Sₙ = a(1 - rⁿ)/(1 - r). This is a classic proof you should be able to reproduce.
Edexcel 可能会要求你证明几何级数的求和公式。标准证明如下：设 Sₙ = a + ar + ar² + ... + arⁿ⁻¹。两边同乘 r：rSₙ = ar + ar² + ar³ + ... + arⁿ。用第一式减去第二式：Sₙ - rSₙ = a - arⁿ → Sₙ(1 - r) = a(1 - rⁿ)。因此，Sₙ = a(1 - rⁿ)/(1 - r)。这是一个你应该能够默写的经典证明。
This proof works for any r ≠ 1 and emphasizes the algebraic manipulation often tested in exams. Remember that subtraction eliminates all intermediate terms.
该证明适用于任何 r ≠ 1，并强调了考试中常考的代数运算。记住，相减会消去所有中间项。
8. Compound Interest and Growth | 复利与增长
ation of geometric sequences. If a principal P is invested at an annual interest rate of i% compounded annually, the amount after t years is A = P(1 + i/100)t amounts year by year: P, P(1+r), P(1+r)², ..., with r = i/100, the common ratio is (1+r).
复利是几何数列的直接应用。如果本金 P 以年利率 i% 按年复利，t 年后的金额为 A = P(1 + i/100)t。如果将第 0 年视为本金，这就是第 (t+1) 项。逐年金额序列为 P, P(1+r), P(1+r)², ...，其中 r = i/100，公比为 (1+r)。
Depreciation problems work similarly: a car loses 15% The value after n years is Vₙ = V₀ × (0.85)ⁿ. This is a geometric sequence with r = 0.85.
折旧问题类似：一辆车每年贬值 15%。n 年后的价值为 Vₙ = V₀ × (0.85)ⁿ。这是一个公比 r = 0.85 的几何数列。
9. Recurrence Relations | 递推关系
alculators or setting up iterative processes. In Edexcel pure maths, recursive definitions often appear in sequences and series questions linking to geometric progressions.
几何数列也可以用递推关系定义：u₁ = a，对于 k ≥ 1 有 uₖ₊₁ = r × uₖ。这种形式在使用计算器或设置迭代过程时很有用。在 Edexcel 纯数学中，递推定义常出现在与等比数列相关的序列和级数问题中。
derstanding. For example, a sequence defined by uₙ₊₁ = 2uₙ, u₁ = 5 immediately tells us it's geometric with a = 5, r = 2.
能够在显式（通项）和递推形式之间切换表明更深的理解。例如，由 uₙ₊₁ = 2uₙ, u₁ = 5 定义的序列立即告诉我们它是首项为 5、公比为 2 的几何数列。
10. Identifying Geometric Sequences | 识别几何数列
n consecutive terms is constant: u₂/u₁ = u₃/u₂ = u₄/u₃ = ... = r. However, be careful with sequences that may have a zero term; if a term is zero, the ratio is undefined unless all terms are zero (degenerate).
常见的考题是判断给定数列是否为几何数列，若是，则求出公比。条件是相邻项的比值恒定：u₂/u₁ = u₃/u₂ = u₄/u₃ = ... = r。但是，要注意可能含有零项的数列；如果某一项为零，除非所有项均为零（退化情况），否则比值无定义。
Also, sometimes a sequence can be geometric only from a certain index onwards. For example, 1, 2, 3, 6, 12, ... the first two terms do not follow a geometric pattern, but from the third term it might. Always check the definition carefully.
此外，有时数列可能仅从某一指标开始才是几何数列。例如，1, 2, 3, 6,
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