IB Math: Geometric Sequences Key Points & Exam Questions | IB 数学：等比数列要点与题型

📚 IB Math: Geometric Sequences Key Points & Exam Questions | IB 数学：等比数列要点与题型

A geometric sequence is one of the most important topics in the IB Math curriculum. It appears in both Analysis & Approaches (AA) and Applications & Interpretation (AI), often in paper 1 and paper 2 questions, as well as in the internal assessment.

等比数列是 IB 数学课程中最重要的内容之一。它出现在分析与方法（AA）和应用与解释（AI）两门课程中，常见于卷一、卷二以及内部评估中。

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1. Definition of a Geometric Sequence | 等比数列的定义

A geometric sequence is a sequence in which each term after the first is obtained by multiplying the previous term by a fixed non-zero number called the common ratio, denoted by r.

等比数列是指从第二项起，每一项与它前一项的比值都等于同一个非零常数，这个常数称为公比，记作 r。

For example, 2, 6, 18, 54, … is a geometric sequence with first term u₁ = 2 and common ratio r = 3.

例如，2, 6, 18, 54, … 是首项 u₁ = 2、公比 r = 3 的等比数列。

We usually write the first term as u₁ or a. The common ratio can be found by dividing any term by the previous term: r = uₙ₊₁ / uₙ.

我们通常将首项写作 u₁ 或 a。公比可以通过任意一项除以前一项得到：r = uₙ₊₁ / uₙ。

It is essential to remember that r ≠ 0. If r = 1, the sequence is constant, for example 5, 5, 5, 5, … .

必须记住 r ≠ 0。若 r = 1，则数列为常数列，例如 5, 5, 5, 5, …。

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2. The General Term (nth Term) | 通项公式

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

等比数列的通项公式为：

uₙ = u₁ × r⁽ⁿ⁻¹⁾

where u₁ is the first term, r is the common ratio, and n is the term number.

其中 u₁ 是首项，r 是公比，n 是项数。

For example, for the sequence 3, 6, 12, 24, … , the nth term is uₙ = 3 × 2⁽ⁿ⁻¹⁾.

例如，对于数列 3, 6, 12, 24, …，其通项为 uₙ = 3 × 2⁽ⁿ⁻¹⁾。

To find a specific term, simply substitute the value of n. For instance, u₅ = 3 × 2⁴ = 48.

要求某一项，只需代入 n 的值。例如 u₅ = 3 × 2⁴ = 48。

In IB exams, you may be given two terms and asked to find the first term and the common ratio. This creates two equations that can be solved simultaneously, often with the use of logarithms or by taking ratios.

在 IB 考试中，可能会给出两项，让你求首项和公比。这需要建立两个方程联立求解，通常借助对数或两式相除。

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3. Finding the Common Ratio | 求公比的方法

The common ratio r can be found in several ways. The most direct method is:

公比 r 有多种求法。最直接的方法是：

r = uₙ₊₁ / uₙ

If you know two non-consecutive terms uₘ and uₙ (with m < n), then:

若已知两项 uₘ 和 uₙ（其中 m < n），则：

uₙ = uₘ × r⁽ⁿ⁻ᵐ⁾

For example, if u₃ = 8 and u₆ = 64, then 64 = 8 × r³, so r³ = 8, giving r = 2.

例如，若 u₃ = 8 且 u₆ = 64，则 64 = 8 × r³，所以 r³ = 8，得 r = 2。

Be careful: when the power is even, there may be two possible values for r, one positive and one negative. Always check whether the sequence is defined to be increasing or decreasing.

注意：当幂为偶数时，r 可能有两个值，一正一负。务必检查题目是否说明数列递增或递减。

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4. Sum of the First n Terms | 前 n 项和公式

The sum of the first n terms of a geometric sequence is given by:

等比数列前 n 项和公式为：

Sₙ = u₁(1 − rⁿ) / (1 − r) , r ≠ 1

An equivalent form, often useful when r > 1, is:

当 r > 1 时，常用等价形式：

Sₙ = u₁(rⁿ − 1) / (r − 1)

Both formulas are given in the IB formula booklet, but you must know when to apply each one.

这两个公式都在 IB 公式手册中给出，但你必须知道何时使用哪一个。

For example, for 2, 6, 18, 54, the sum of the first 4 terms is S₄ = 2(1 − 3⁴) / (1 − 3) = 2(1 − 81) / (−2) = 80.

例如，对于 2, 6, 18, 54，前 4 项和为 S₄ = 2(1 − 3⁴) / (1 − 3) = 2(1 − 81) / (−2) = 80。

A common exam question is to find the sum of terms from position p to position q. In this case, calculate Sq − Sₚ₋₁.

常见考题是求从第 p 项到第 q 项的和。此时计算 Sq − Sₚ₋₁ 即可。

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5. Infinite Geometric Series | 无穷等比数列的和

An infinite geometric series converges only when the common ratio satisfies |r| < 1.

无穷等比数列仅在公比满足 |r| < 1 时收敛。

The sum to infinity is given by:

无穷和公式为：

S∞ = u₁ / (1 − r) , |r| < 1

If |r| ≥ 1, the series does not converge and the sum to infinity does not exist.

若 |r| ≥ 1，级数不收敛，因此无穷和不存在。

For example, the series 100 + 50 + 25 + 12.5 + … has u₁ = 100 and r = 1/2. Its sum to infinity is 100 / (1 − 1/2) = 200.

例如，级数 100 + 50 + 25 + 12.5 + … 中 u₁ = 100，r = 1/2。其无穷和为 100 / (1 − 1/2) = 200。

This concept appears frequently in IB, especially in questions involving recurring decimals, fractals, or geometric settings.

这一概念在 IB 中经常出现，尤其是循环小数、分形或几何背景的题目中。

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6. Recurring Decimals and Geometric Series | 循环小数与等比数列

One classic IB application is writing a recurring decimal as an exact fraction using an infinite geometric series.

IB 中一个经典应用是利用无穷等比数列将循环小数写为精确分数。

For example, 0.3333… = 0.3 + 0.03 + 0.003 + … , with u₁ = 0.3 and r = 0.1.

例如，0.3333… = 0.3 + 0.03 + 0.003 + …，其中 u₁ = 0.3，r = 0.1。

Using the infinite sum formula: S∞ = 0.3 / (1 − 0.1) = 0.3 / 0.9 = 1/3.

利用无穷和公式：S∞ = 0.3 / (1 − 0.1) = 0.3 / 0.9 = 1/3。

For a decimal like 0.272727…, the terms are 0.27 + 0.0027 + 0.000027 + … , giving S∞ = 0.27 / (1 − 0.01) = 27/99 = 3/11.

对于 0.272727…，各项为 0.27 + 0.0027 + 0.000027 + …，得 S∞ = 0.27 / (1 − 0.01) = 27/99 = 3/11。

When you have a mixed repeating decimal such as 0.125555…, separate the non-repeating part first, then treat the repeating tail as an infinite geometric series.

对于混循环小数如 0.125555…，先分离不循环部分，再将循环部分视为无穷等比数列。

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7. Compound Interest and Financial Applications | 复利与金融应用

In IB AI and Applications, geometric sequences are heavily used in compound interest problems. The formula is:

在 IB AI 与应用课程中，等比数列大量用于复利问题。公式为：

FV = PV × (1 + i)ⁿ

where FV is the future value, PV is the present value, i is the interest rate per period, and n is the number of periods.

其中 FV 是终值，PV 是现值，i 是每期利率，n 是期数。

If interest is compounded monthly, divide the annual rate by 12 and multiply the number of years by 12.

若按月复利，则将年利率除以 12，并将年数乘以 12。

For example, if you invest 1000 at an annual interest rate of 5% compounded annually, after 10 years the value is 1000 × (1.05)¹⁰ ≈ 1628.89.

例如，若以年利率 5% 每年复利投资 1000，10 年后的值为 1000 × (1.05)¹⁰ ≈ 1628.89。

Remember that each period the multiplier is (1 + i), so the value follows a geometric sequence with common ratio r = 1 + i.

记住每一期的乘数为 (1 + i)，因此数值构成公比 r = 1 + i 的等比数列。

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8. Geometric Mean and Problem Solving | 等比中项与解题技巧

For three consecutive terms of a geometric sequence, the middle term is the geometric mean of the other two:

在等比数列中，连续三项满足中间一项是另外两项的等比中项：

b² = a × c

This property is often used to find an unknown term when surrounded by known terms.

这一性质常被用来求已知两项中间的那一项。

For example, if x, 6, 12 are consecutive terms of a geometric sequence, then 6² = 12x, so 36 = 12x, giving x = 3.

例如，若 x, 6, 12 是等比数列的连续三项，则 6² = 12x，即 36 = 12x，所以 x = 3。

However, note that x = 3 gives a positive ratio. If x = −3, then 6 / (−3) = −2 and 12 / 6 = 2, which is not a common ratio. Always verify consistency across all terms.

但注意，x = 3 给出正公比。若 x = −3，则 6 / (−3) = −2 而 12 / 6 = 2，公比不一致。务必验证所有项是否满足同一公比。

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9. Common Mistakes and Exam Warnings | 常见错误与考试警示

• uₙ = u₁rⁿ is wrong. The exponent must be n − 1, not n.
• uₙ = u₁rⁿ 是错误的。指数必须是 n − 1，不是 n。
• Forgetting the condition |r| < 1 before using the infinite sum formula.
• 使用无穷和公式前忘记条件 |r| < 1。
• Using the sum formula when r = 1, which leads to division by zero.
• 当 r = 1 时使用求和公式，会导致除以零。
• Confusing geometric sequences with arithmetic sequences, especially in word problems.
• 在应用题中混淆等比数列与等差数列。
• Misreading “sum from term 3 to term 8” and directly substituting n = 8 without subtracting the first two terms.
• 误读“第 3 项到第 8 项之和”，直接代入 n = 8 而不减去前两项。

In IB, marks are often deducted for missing justifications. Always state the common ratio and rewrite the formula before substituting values.

在 IB 中，缺少必要说明通常会被扣分。务必写出公比并先写公式再代值。

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10. Worked Exam-Style Example | 典型真题例题

Question: The third term of a geometric sequence is 18 and the sixth term is 486. Find the first term, the common ratio, and the sum of the first 8 terms.

题目：已知等比数列的第三项为 18，第六项为 486。求首项、公比以及前 8 项之和。

Step 1: Write two equations:

第一步：写出两个方程：

u₃ = u₁r² = 18 , u₆ = u₁r⁵ = 486

Step 2: Divide the second equation by the first:

第二步：将第二个方程除以第一个方程：

r³ = 486 / 18 = 27

Therefore r = 3. Since all terms are positive, we take the positive root.

因此 r = 3。由于所有项为正，取正根。

Step 3: Substitute r = 3 into u₁r² = 18:

第三步：将 r = 3 代入 u₁r² = 18：

u₁ × 9 = 18 , u₁ = 2

Step 4: Compute the sum of the first 8 terms:

第四步：计算前 8 项之和：

S₈ = 2(3⁸ − 1) / (3 − 1) = 2(6561 − 1) / 2 = 6560

This example shows the standard two-equation method that appears frequently in IB papers.

此例展示了 IB 试卷中常见的两方程联立求解法。

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11. Tips for IB Exam Success | IB 考试高分建议

Always write down the general formula before substituting numbers. This helps the examiner follow your reasoning and earns method marks.

每次代入数值前先写出通项公式或求和公式。这有助于考官理解你的思路，并获得方法分。

When using your GDC in the AI paper, make sure you know how to enter sequences and sums efficiently. However, for AA paper 1, you must solve by hand and present exact answers.

在 AI 卷中使用图形计算器时，请确保能高效输入数列与求和。但在 AA 卷一，你必须手算并给出精确答案。

Practice converting between exponential and logarithmic form, since geometric sequence problems often require solving rⁿ = k using logarithms.

练习指数形式与对数形式的互化，因为等比数列问题经常需要通过取对数求解 rⁿ = k。

Finally, always check whether the final answer makes sense in context: negative sums, enormous ratios, or inconsistent terms are warning signs.

最后，始终检查最终答案是否符合实际背景：负和、超大公比或前后不一致的项都是危险信号。

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12. Summary | 总结

Concept | 概念	Formula | 公式
nth term | 通项	uₙ = u₁r⁽ⁿ⁻¹⁾
Sum of n terms | 前 n 项和	Sₙ = u₁(1 − rⁿ) / (1 − r)
Sum to infinity | 无穷和	S∞ = u₁ / (1 − r) , |r| < 1
Geometric mean | 等比中项	b² = a × c

Master these key formulas and practise exam-style questions. Geometric sequences are a reliable source of marks once you are comfortable with the algebra and applications.

掌握这些核心公式并练习真题题型。一旦你熟悉了相关代数和应用，等比数列就是稳定的得分点。

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