Geometric Sequences | 等比数列

📚 Geometric Sequences | 等比数列

In IB Mathematics, geometric sequences form a fundamental building block for understanding exponential growth, financial mathematics, and infinite processes. Whether you are studying Analysis & Approaches or Applications & Interpretation, you will encounter geometric sequences in topics ranging from series summation to modelling real-world phenomena. This article offers a comprehensive revision of geometric sequences, complete with definitions, key formulas, worked examples, and exam-focused advice.

在 IB 数学中，等比数列是理解指数增长、金融数学和无穷过程的基础内容。无论你学习的是分析与方法还是应用与解释，都会在级数求和和现实世界建模等主题中遇到等比数列。本文为你提供等比数列的全面复习，包括定义、关键公式、解题示例以及针对 IB 考试的实用建议。

1. Definition and Common Ratio | 定义与公比

A geometric sequence is an ordered list of numbers in which each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio, usually denoted by r. For example, the sequence 2, 6, 18, 54, ... has r e preceding one. If the common ratio is negative, the terms will alternate in sign. The first term is conventionally written as u₁ or a.

等比数列是一组有序的数，其中从第二项起，每一项都等于前一项乘以一个固定的非零常数，这个常数称为公比，通常记作 r。例如，数列 2, 6, 18, 54, … 的公比 r = 3，因为每一项都是前一项的 3 倍。如果公比为负数，数列的项会正负交替。首项通常写作 u₁ 或 a。

2. The nth Term Formula | 第 n 项公式

Given the first term u₁ and common ratio r, the nth term of a geometric sequence is expressed as

u_n = u₁ r^{n − 1}

This powerful formula allows you to find any term directly without calculating all the preceding terms. For instance, if the first term is 5 and r = 2, the 9th term is 5 × 2⁸ = 1280. Be careful with the exponent: it is (n − 1), not n.

已知首项 u₁ 和公比 r，等比数列的第 n 项可表示为

u_n = u₁ r^{n − 1}

这个强大的公式能让你直接求出任意项，而无需逐一计算前面的项。例如，若首项为 5，公比 r = 2，则第 9 项为 5 × 2⁸ = 1280。注意指数是 (n − 1)，而不是 n。

3. Finding Terms and the Common Ratio | 求项与公比

To find the common ratio when two consecutive terms are known, simply divide a term by its predecessor: r = u_k+1 / u_k. If the terms are not consecutive, you can still determine r by solving an equation. For example, if the 3rd term is 18 and the 6th term is 486, then u₆ = u₃ r³, leading to 486 = 18 r³, hence r³ = 27 and r ratio is constant for all pairs of consecutive terms to confirm the sequence is geometric.

当已知连续两项时，只需用后项除以前项便可求出公比：r = u_k+1 / u_k。如果两项不连续，也可以通过解方程来确定 r。例如，若第 3 项为 18，第 6 项为 486，则由 u₆ = u₃ r³ 得 486 = 18 r³，即 r³ = 27，所以 r = 3。务必检查所有连续项对的比值是否恒定，以确认数列是等比数列。

4. Geometric Mean | 等比中项

The geometric mean of two numbers a and c is the number b such that a, b, c form a three‑term geometric sequence. This gives the relationship b² = a × c, so b = ±√a c. For positive numbers the positive root is taken. The concept is particularly useful in IB problems that ask you to insert one or more geometric means between two given numbers, thus building a longer geometric sequence.

两个数 a 与 c 的等比中项是指满足 a, b, c 成三项等比数列的数 b。由此得到关系式 b² = a × c，即 b = ±√a c。对正数通常取正平方根。这一概念在 IB 题目中尤为有用，例如要求你在两个给定数之间插入一个或多个等比中项，从而构造一个更长的等比数列。

5. Sum of a Finite Geometric Series | 有限等比级数的和

The sum of the first n terms of a geometric sequence is called a finite geometric series. The sum S_n can be calculated using the formula

S_n = u₁ (1 − rⁿ) / (1 − r)   for   r ≠ 1

If r = 1, all terms are equal and S_n = n × u₁. An alternative form, S_n = u₁ (rⁿ − 1) / (r − 1), can be convenient when r > 1. Always verify which version you are using to avoid sign errors.

等比数列前 n 项的和称为有限等比级数。其和 S_n 的公式为

S_n = u₁ (1 − rⁿ) / (1 − r)   当   r ≠ 1

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若 r = 1，则所有项都相等，S_n = n × u₁。另一种等价形式 S_n = u₁ (rⁿ − 1) / (r − 1) 在 r > 1 时更为方便。请务必确认你所使用的版本，以避免符号错误。

6. Sum to Infinity of a Geometric Series | 无穷等比级数的和

When the absolute value of the common ratio is less than 1, |r| < 1, the terms of the geometric sequence become progressively smaller, and the infinite series converges to a finite limit. This sum to infinity is given by

S_∞ = u₁ / (1 − r)   for   |r| < 1

If |r| sum exists. IB exam questions frequently test this condition, asking you first to prove convergence before calculating the sum.

当公比的绝对值小于 1，即 |r| < 1 时，等比数列的各项将逐项减小，无穷级数收敛到一个有限值。这个无穷级数的和公式为

S_∞ = u₁ / (1 − r)   当   |r| < 1

若 |r| ≥ 1，级数发散，不存在有限和。IB 试题经常考查这一条件，要求你先证明级数收敛，然后再求和。

7. Sigma Notation for Geometric Series | 等比级数的求和符号

Geometric series can be compactly written using sigma notation: ∑ⁿ_k=1 u₁ r^k−1. When the index begins at 0, the sum of the first n terms is ∑ⁿ⁻¹_k=0 u₁ r^k. Familiarity with different starting indices is essential because IB questions may present series in several formats. You can convert between forms by factoring out common powers or shifting the index.

等比级数可用求和符号简洁地表示：∑ⁿ_k=1 u₁ r^k−1。当指标从 0 开始时，前 n 项的和为 ∑ⁿ⁻¹_k=0 u₁ r^k。熟悉不同的起始指标至关重要，因为 IB 试题可能以多种形式呈现级数。你可以通过提取公共幂或平移指标在两种形式间转换。

8. Applications: Growth and Decay | 应用：增长与衰减

Geometric sequences model a wide range of real‑world situations. Compound interest, where an investment grows by a fixed percentage each period, follows the rule A_n = P(1 + i)ⁿ, with P as the principal and i the interest rate per period. Population growth with a constant growth rate, radioactive decay, and depreciation of assets can similarly be described by geometric sequences, with the common ratio reflecting the growth factor (>1) or decay factor (0 < r < 1). IB problems often integrate these contexts, requiring you to identify u₁ and r, then use sequence formulas to answer further questions.

等比数列可用于模拟多种现实情境。复利投资每期按固定百分比增长，遵循公式 A_n = P(1 + i)ⁿ，其中 P 为本金，i 为每期利率。具有恒定增长率的人口增长、放射性衰变以及资产折旧同样可以用等比数列描述，公比反映增长因子（>1）或衰减因子（0 < r < 1）。IB 题目常结合这些场景，要求你先识别 u₁ 和 r，再运用数列公式解答后续问题。

9. Using Logarithms to Solve for n | 使用对数求解 n

Many geometric sequence problems require finding the term number n when the value of the nth term is given. Starting from u_n = u₁ rⁿ⁻¹ range to isolate the exponential part and then take logarithms of both sides (usually base‑10 or natural log). For example, 3 × 2ⁿ⁻¹ = 1536 gives 2ⁿ⁻¹ = 512. Since 512 = 2⁹, n − 1 = 9, but for non‑integer results, logarithms are indispensable: n = log(u_n/u₁) / log(r) + 1.

许多等比数列问题需要在已知第 n 项值时求出项数 n。从 u_n = u₁ rⁿ⁻¹ 出发，分离指数部分，然后两边取对数（常用对数或自然对数）。例如 3 × 2ⁿ⁻¹ = 1536 可化为 2ⁿ⁻¹ = 512。由于 512 = 2⁹，得 n − 1 = 9；但当结果不是整数时，对数就必不可少：n = log(u_n/u₁) / log(r) + 1。

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

• Always confirm |r| < 1 before using the sum to infinity formula.
• Be mindful of the term number: u_n = u₁ rⁿ⁻¹, not u₁ rⁿ.
• Use the correct form of the sum formula based on the value of r to avoid sign mistakes.
• When solving for n with logarithms, remember to add 1 after dividing the logs if your formula uses n−1.
• In applications, distinguish between the initial term (often u₀) and the first term u₁.
• Show all working clearly; IB awards marks for method, even if the final answer contains a small numerical error.

• 使用无穷级数和公式前，务必确认 |r| < 1。
• 注意项数序号：通项公式为 u_n = u₁ rⁿ⁻¹，而非 u₁ rⁿ。
• 根据 r 的值选用正确的求和公式形式，避免符号错误。
• 用对数求 n 时，若公式中使用的是 n−1，则在除以对数后别忘了加 1。
• 在应用题中，要区分初始项（常为 u₀）与第一项 u₁。
• 清晰写出所有解题步骤；IB 的评分注重方法，即使最后答案出现小数值错误也能得到部分分数。

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