Exercise 1D.1: Geometric Sequences | 练习1D.1：等比数列

📚 Exercise 1D.1: Geometric Sequences | 练习1D.1：等比数列

In IB Mathematics, Exercise 1D.1 often marks your first deep dive into geometric sequences – an essential algebraic tool for modelling growth, decay, and many real-world patterns. This article unpacks every core concept, from the common ratio to the general term and beyond, so you can tackle both SL and HL questions with confidence.

在 IB 数学中，练习 1D.1 通常是你第一次深入探索等比数列的地方——它是描述增长、衰减和许多现实模式的重要代数工具。本文拆解了从公比到通项公式的每一个核心概念，帮助你自信应对 SL 和 HL 的问题。

1. What Is a Geometric Sequence? | 什么是等比数列？

A geometric sequence (also called a geometric progression) is an ordered list of numbers in which each term after the first is obtained by multiplying the preceding term by a fixed, non-zero constant. This constant multiplier is called the common ratio, denoted by r.

等比数列（又称几何数列）是一串有序的数字，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。这个固定的乘数称为公比，记作 r。

For example, 3, 12, 48, 192, … is a geometric sequence with first term u₁ = 3 and common ratio r = 4, because 12÷3 = 4, 48÷12 = 4, and so on.

例如，3, 12, 48, 192, … 是一个等比数列，首项 u₁ = 3，公比 r = 4，因为 12÷3 = 4，48÷12 = 4，依此类推。

Note that the common ratio can be negative (causing alternating signs) or a fraction between -1 and 1 (causing the terms to shrink in magnitude). The only restriction is that r ≠ 0.

注意，公比可以是负数（导致正负交替出现），也可以是介于 –1 和 1 之间的分数（使项的绝对值逐渐减小）。唯一的限制是 r ≠ 0。

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2. Identifying the Common Ratio | 识别公比

To find the common ratio, simply divide any term by the term immediately before it: r = u₂/u₁ = u₃/u₂ = u₄/u₃ …

要找公比，只需将任意一项除以其前一项：r = u₂/u₁ = u₃/u₂ = u₄/u₃ …

Consider the sequence –2, 6, –18, 54, … . Here, r = 6 ÷ (–2) = –3. The negative ratio makes the terms alternate in sign, which is a common feature in IB exam questions.

考虑数列 –2, 6, –18, 54, … 。这里 r = 6 ÷ (–2) = –3。负公比使项的符号交替出现，这是 IB 考试题中的常见特征。

If the terms are given as fractions or decimals, the same rule applies: for 0.1, 0.01, 0.001, … the ratio is r = 0.01 ÷ 0.1 = 0.1. You can always check consistency by testing several pairs.

如果各项以分数或小数的形式给出，同样的规则也适用：对于 0.1, 0.01, 0.001, … ，公比 r = 0.01 ÷ 0.1 = 0.1。你可以通过检验多组相邻项来确认一致性。

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3. The General Term Formula | 通项公式

The nth term of a geometric sequence is expressed by the formula:

等比数列的第 n 项（通项）用以下公式表示：

uₙ = u₁ × rⁿ⁻¹

where u₁ is the first term, r is the common ratio, and n is the term number (n ≥ 1). This formula is on your IB formula booklet, but understanding how to manipulate it is key.

其中 u₁ 是首项，r 是公比，n 是项数（n ≥ 1）。这个公式已在 IB 公式手册中给出，但理解如何灵活运用它是关键。

Derivation: Starting from u₁, the second term is u₂ = u₁ × r, the third is u₃ = u₁ × r², and by induction, the nth term adds one more factor of r than the previous, giving exactly rⁿ⁻¹.

推导：从 u₁ 出发，第二项 u₂ = u₁ × r，第三项 u₃ = u₁ × r²，依此类推，通过归纳可知第 n 项比首项多乘了 (n–1) 个 r，即 rⁿ⁻¹。

For example, the 10th term of the sequence with u₁ = 5 and r = 2 is u₁₀ = 5 × 2⁹ = 5 × 512 = 2560.

例如，首项 u₁ = 5、公比 r = 2 的数列，第 10 项为 u₁₀ = 5 × 2⁹ = 5 × 512 = 2560。

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4. Finding a Specific Term | 求特定项

IB questions often ask you to calculate a distant term without listing all intermediate ones. You simply substitute the known values into uₙ = u₁ × rⁿ⁻¹.

IB 题目常常要求你直接计算某个靠后的项，而不必列出所有中间项。你只需将已知量代入 uₙ = u₁ × rⁿ⁻¹ 即可。

Example: The 1st term of a geometric sequence is 7 and r = 3. Find the 6th term.

Solution: u₆ = 7 × 3⁶⁻¹ = 7 × 3⁵ = 7 × 243 = 1701.

示例：某等比数列的首项为 7，公比 r = 3。求第 6 项。

解：u₆ = 7 × 3⁶⁻¹ = 7 × 3⁵ = 7 × 243 = 1701。

When working with a negative ratio, remember to handle the sign carefully. For u₁ = 4, r = –2, the 7th term is u₇ = 4 × (–2)⁶ = 4 × 64 = 256 (even power, so positive).

当公比为负数时，务必谨慎处理符号。对于 u₁ = 4，r = –2，第 7 项为 u₇ = 4 × (–2)⁶ = 4 × 64 = 256（指数为偶数，结果为正）。

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5. Determining the Number of Terms | 确定项数

Another typical exam task is to find how many terms a given geometric sequence contains, especially when the last term is known.

另一类典型考题是已知首项、公比和末项，求该等比数列共有多少项。

Example: How many terms are in the sequence 3, 6, 12, …, 1536? Here u₁ = 3, r = 2, and the final term uₙ = 1536. Set up the equation: 3 × 2ⁿ⁻¹ = 1536 → 2ⁿ⁻¹ = 512 → 2ⁿ⁻¹ = 2⁹ → n – 1 = 9 → n = 10.

示例：数列 3, 6, 12, …, 1536 有多少项？这里 u₁ = 3，r = 2，末项 uₙ = 1536。建立方程：3 × 2ⁿ⁻¹ = 1536 → 2ⁿ⁻¹ = 512 → 2ⁿ⁻¹ = 2⁹ → n – 1 = 9 → n = 10。

If the ratio is a fraction, the same logic applies; just be prepared to solve using logarithms when the numbers are not neat powers. The IB allows use of the GDC, so you can also use the solver or graph intersection.

如果公比是分数，同样的逻辑依然适用；当数字不是整齐的幂时，要做好用对数求解的准备。IB 考试允许使用图形计算器，因此你也可以使用方程求解器或图像交点功能。

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6. The Geometric Mean | 等比中项

When three numbers a, b, c are consecutive terms in a geometric sequence, b is called the geometric mean of a and c, satisfying b² = a × c.

当三个数 a, b, c 构成等比数列的连续三项时，b 称为 a 与 c 的等比中项，且满足 b² = a × c。

For instance, 4, 12, 36 are in geometric progression because 12² = 144 and 4 × 36 = 144. The geometric mean is always the square root of the product of the two outer terms, though sign must be chosen according to context.

例如，4, 12, 36 构成等比数列，因为 12² = 144 且 4 × 36 = 144。等比中项总是两端项乘积的平方根，但需要根据题意选择合适的符号。

This concept is frequently tested in IB alongside the arithmetic mean, so be careful to apply the correct relation. For any two positive numbers, the geometric mean is less than or equal to the arithmetic mean.

在 IB 考试中，这一概念常与等差中项一同考查，因此要注意使用正确的关系式。对于任意两个正数，等比中项总是小于或等于它们的等差中项。

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7. Recurrence Form of a Geometric Sequence | 等比数列的递推形式

A geometric sequence can also be defined recursively: uₙ₊₁ = r × uₙ, with the initial term u₁ given. This highlights the multiplicative nature of the sequence.

等比数列也可以用递推关系来定义：uₙ₊₁ = r × uₙ，并给出首项 u₁。这突显了数列的乘法特性。

For example, if u₁ = 10 and r = 1.5, then the recursion uᵢ₊₁ = 1.5 × uᵢ generates the sequence 10, 15, 22.5, 33.75, … . Recurrence is particularly useful in programming or iterative modelling.

例如，若 u₁ = 10，r = 1.5，那么递推式 uᵢ₊₁ = 1.5 × uᵢ 将生成数列 10, 15, 22.5, 33.75, … 。递推在编程或迭代建模中特别有用。

IB problems may ask you to write a recurrence formula from a description or to find r when given two consecutive terms. Always remember that r = uₙ₊₁ / uₙ.

IB 题目可能要求你根据文字描述写出递推公式，或根据相邻两项求出 r。时刻记住 r = uₙ₊₁ / uₙ。

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8. Applications: Compound Interest | 应用：复利计算

One of the most practical uses of geometric sequences is in compound interest. When an amount P is invested at an annual interest rate r% compounded annually, the value after n years follows a geometric progression:

等比数列最实际的应用之一是复利计算。当本金 P 以年利率 r% 每年复利一次时，n 年后的终值遵循等比数列：

A = P × (1 + i)ⁿ

where i = r/100 is the decimal interest rate per compounding period. This is exactly the general term of a geometric sequence with u₁ = P and common ratio (1 + i).

其中 i = r/100 是每个计息期的十进制利率。这正是一个首项 u₁ = P、公比为 (1 + i) 的等比数列的通项。

Example: $2000 invested at 5% per annum compounded annually becomes 2000 × (1.05)⁵ ≈ $2552.56 after 5 years. If compounded monthly, the common ratio per month would be 1 + 0.05/12, and the exponent becomes 12n.

示例：2000 美元按年利率 5% 每年复利一次，5 年后将变为 2000 × (1.05)⁵ ≈ 2552.56 美元。若按月复利，则每月的公比为 1 + 0.05/12，指数变为 12n。

Recognising the underlying geometric structure allows you to apply the sequence formula directly, avoiding confusion with exponential function notation.

识别出潜在的等比数列结构，你就能直接套用数列公式，从而避免与指数函数符号混淆。

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9. Exponential Decay and Half-Life | 指数衰减与半衰期

A geometric sequence with 0 < r < 1 models exponential decay. This appears in depreciation, radioactive decay, and cooling. The half-life is the time taken for a quantity to halve, which corresponds to finding n such that rⁿ = ½.

公比满足 0 < r < 1 的等比数列可以模拟指数衰减，常见于折旧、放射性衰变和冷却过程中。半衰期是指数量减半所需的时间，也就是找到满足 rⁿ = ½ 的 n。

For a radioactive substance with a decay factor of 0.9 per year, the remaining mass after n years is uₙ = M₀ × 0.9ⁿ⁻¹ if we count from year 1. To find when it halves, solve 0.9ⁿ⁻¹ = 0.5 using logarithms.

对于每年衰减因子为 0.9 的放射性物质，若从第 1 年算起，剩余质量可表示为 uₙ = M₀ × 0.9ⁿ⁻¹。要求何时减半，需用对数解方程 0.9ⁿ⁻¹ = 0.5。

Setting up an IB problem like this: “A car loses 15% of its value each year. If it was bought for $30,000, write a sequence for its value and find when it falls below $5,000.” The common ratio r = 0.85, and you solve 30000 × 0.85ⁿ⁻¹ < 5000.

IB 题目的典型设置如：“一辆汽车每年贬值 15%。如果买入价为 30000 美元，写出其价值数列，并求何时价值低于 5000 美元。”此时公比 r = 0.85，需要解 30000 × 0.85ⁿ⁻¹ < 5000。

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10. IB Exam Tips for Geometric Sequences | IB 等比数列考试技巧

Here are some targeted tips to help you avoid common pitfalls in Exercise 1D.1 and beyond:

以下是一些针对性的建议，帮助你避开练习 1D.1 及后续内容中的常见陷阱：

• Check the sign of r. If terms alternate in sign, r is negative. Use parentheses when raising to a power: (–3)² = 9, not –3² = –9. (英文)
  检查 r 的符号。若各项符号交替，则 r 为负数。乘方时要加括号：(–3)² = 9，而 –3² = –9。(中文)
• Use the formula booklet wisely. You are given uₙ = u₁ rⁿ⁻¹ but do not confuse it with the arithmetic formula. (英文)
  善用公式手册。手册中给出了 uₙ = u₁ rⁿ⁻¹，但不要与等差数列公式混淆。(中文)
• Logarithms are your friend. When solving for n in uₙ = k, take logs: n⁻¹ = log(k/u₁) / log(r), then add 1. (英文)
  对数是你可靠的工具。当需要从 uₙ = k 中解出 n 时，两边取对数：n⁻¹ = log(k/u₁) / log(r)，再加 1。(中文)
• Geometric vs arithmetic mean. For a, b, c in geometric progression, b² = a c; for arithmetic, b = (a + c)/2. (英文)
  区分等比中项与等差中项。对于等比 a, b, c，有 b² = a c；对于等差，有 b = (a + c)/2。(中文)
• Watch the index offset. In recurrence uₙ₊₁ = r uₙ with u₁ = a, the nth term is indeed a × rⁿ⁻¹, not a × rⁿ. (英文)
  注意指数偏移。在递推 uₙ₊₁ = r uₙ 且 u₁ = a 的情况下，第 n 项确实是 a × rⁿ⁻¹，而不是 a × rⁿ。(中文)

A quick reference table summarises key information for geometric sequences:

以下速查表总结了等比数列的关键信息：

Element	Formula/Description
Common ratio (r)	r = uₙ₊₁ / uₙ, r ≠ 0
General term (uₙ)	uₙ = u₁ × rⁿ⁻¹
Geometric mean (b)	b = ±√(a c) for a, b, c in GP
Recurrence relation	uₙ₊₁ = r × uₙ, u₁ given
Condition for growth/decay	|r| > 1 ⇒ growth; |r| < 1 ⇒ decay; r = 1 ⇒ constant

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11. From Geometric Sequences to Geometric Series | 从等比数列到等比级数

While Exercise 1D.1 focuses on sequences, the natural extension is the geometric series, which sums the terms of a geometric sequence. The sum of the first n terms is Sₙ = u₁ (1 – rⁿ) / (1 – r) for r ≠ 1. Recognising the sequence properties will make series work much smoother.

虽然练习 1D.1 聚焦于数列，但其自然延伸是等比级数，即等比数列各项之和。前 n 项和为 Sₙ = u₁ (1 – rⁿ) / (1 – r)（r ≠ 1）。掌握数列的特性会让级数学习更加顺畅。

An infinite geometric series converges to a finite sum only when |r| < 1, given by S∞ = u₁ / (1 – r). This is a frequent HL topic, so an early solid grounding in geometric sequences pays off.

无穷等比级数仅在 |r| < 1 时收敛于一个有限和，公式为 S∞ = u₁ / (1 – r)。这是 HL 的常考主题，因此尽早打牢等比数列的基础会令你受益匪浅。

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12. Practice Mindset and Final Advice | 练习心态与最终建议

Mastering Exercise 1D.1 means you are not merely computing terms; you are learning to recognise multiplicative patterns in data, a skill that will serve you in modelling, calculus (exponential functions), and financial mathematics. Practice with varied examples – including fractional and negative ratios – until the general term formula becomes second nature.

掌握练习 1D.1 意味着你不仅是在计算数列的项，更是在学习识别数据中的乘法模式，这一技能会在建模、微积分（指数函数）和金融数学中为你服务。请用各种示例进行练习——包括分数公比和负公比——直到通项公式成为你的第二天性。

Always begin an IB geometric sequence problem by writing down u₁ and r explicitly. Then note what the question asks: a particular term, the number of terms, a missing ratio, or an application. With a structured approach, you’ll turn every question into a simple substitution exercise.

做 IB 等比数列题时，始终从明确写出 u₁ 和 r 开始。然后看清题目要求：是求特定项、项数、缺失的公比，还是实际应用。有了结构化的方法，你就能把每道题都变成一个简单的代入练习。

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