Exercise 1D.2: Geometric Sequences and Applications | 练习 1D.2：等比数列及其应用

📚 Exercise 1D.2: Geometric Sequences and Applications | 练习 1D.2：等比数列及其应用

Welcome to your focused revision on Exercise 1D.2, a key set of practice problems typically covering geometric sequences and their real-world applications in the IB Mathematics course. This guide unpacks the core concepts, formulas, and problem-solving strategies you will encounter, ensuring you can tackle pattern recognition, term calculation, sum evaluation, and applied scenarios with confidence.

欢迎来到练习 1D.2 专项复习，这组习题通常涵盖 IB 数学课程中的等比数列及其实际应用。本指南将梳理核心概念、公式和解题策略，帮助你从容应对模式识别、求项、求和以及应用场景。

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

A geometric sequence is a list of numbers where the ratio between any term (after the first) and the previous term is constant. This constant multiplier is called the common ratio, denoted by ‘r’. The sequence can be finite or infinite, and its behaviour depends heavily on the value of r.

等比数列是一列数字，其中任意一项（第一项之后）与前一项的比值是一个常数。这个常数乘数称为公比，记作 r。数列可以是有限的或无限的，其形态很大程度上取决于 r 的值。

Example: 3, 6, 12, 24, … has r = 2. | 例：3, 6, 12, 24, … 的公比为 2。

You must be able to distinguish a geometric sequence from an arithmetic sequence. In arithmetic, we add or subtract a common difference; in geometric, we multiply by a common ratio. Always check by dividing consecutive terms.

你必须能够区分等比数列和等差数列。等差数列中，我们加减一个公差；等比数列中，我们乘以一个公比。务必通过连续两项相除来检验。

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2. The Common Ratio ‘r’ and Its Effects | 公比 r 及其影响

The common ratio r can be positive, negative, or even a fraction. If |r| > 1, the terms grow in magnitude; if 0 < |r| < 1, the terms decay towards zero. A negative r causes the terms to alternate in sign, creating an oscillating pattern.

公比 r 可以是正数、负数，甚至是分数。如果 |r| > 1，各项绝对值增大；如果 0 < |r| < 1，各项向零衰减。负的 r 会使各项符号交替，产生振荡模式。

To find r, take any term and divide by the previous term: r = u₂/u₁ = u₃/u₂, and so on. In formula-heavy questions, clearly stating the value of r is often the first step toward a solution.

求 r 时，任取一项除以前一项：r = u₂/u₁ = u₃/u₂，以此类推。在公式密集的题目中，清晰写明 r 的值往往是解题的第一步。

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3. The nth Term Formula | 第 n 项公式

The nth term of a geometric sequence is given by the explicit formula below. This formula allows you to jump directly to any term without generating all previous terms, provided you know the first term u₁ and the common ratio r.

等比数列的第 n 项由以下显式公式给出。只要知道首项 u₁ 和公比 r，你就能直接跳到任意一项，无需生成前面所有项。

uₙ = u₁ × rⁿ⁻¹

Here, u₁ is the first term, r is the common ratio, and n is the term number. Be careful: the exponent is (n − 1), not n. A frequent mistake is applying u₁ × rⁿ, which overshoots by one multiplication.

其中 u₁ 为首项，r 为公比，n 为项数。注意：指数为 (n − 1) 而非 n。常见错误是使用 u₁ × rⁿ，这会多乘一次。

When using a graphing calculator (GDC), you can generate terms recursively or use sequence mode with this explicit formula to verify results from Exercise 1D.2.

使用图形计算器（GDC）时，你可以通过递推或序列模式输入该显式公式，来验证练习 1D.2 的结果。

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4. Finding Unknown Terms and the First Term | 求未知项和首项

Many problems in this exercise present two known terms (e.g. u₃ = 18 and u₆ = 486) and ask for the first term and common ratio. Set up two equations using uₙ = u₁ rⁿ⁻¹ and divide them to eliminate u₁. This leaves an equation in r that you can solve, then back-substitute to find u₁.

本练习中许多题目会给出两个已知项（如 u₃ = 18 和 u₆ = 486），要求首项和公比。利用 uₙ = u₁ rⁿ⁻¹ 建立两个方程，将它们相除以消去 u₁，得到关于 r 的方程，解出后代回求出 u₁。

For terms uₘ and uₙ: rⁿ⁻ᵐ = uₙ / uₘ | 对于项 uₘ 和 uₙ：rⁿ⁻ᵐ = uₙ / uₘ

If the sequence is decreasing or alternating, r may be a fraction or negative. Always check whether your r satisfies both original equations and fits the context of the question.

如果数列递减或正负交替，r 可能是分数或负数。务必检查所求 r 是否同时满足两个原方程，是否符合题目背景。

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5. Geometric Series: Sum of the First n Terms | 等比级数：前 n 项求和

A geometric series is the sum of terms from a geometric sequence. The sum of the first n terms, Sₙ, has a compact formula that depends on r. You need to know two versions: one for r ≠ 1, and the special case when r = 1 where the sum is simply n × u₁.

等比级数是等比数列各项的和。前 n 项和 Sₙ 有一个依赖于 r 的紧凑公式。你需要掌握两种形式：一种用于 r ≠ 1；特例 r = 1 时，和就是 n × u₁。

Sₙ = u₁(1 − rⁿ) / (1 − r) for r ≠ 1 | Sₙ = u₁(1 − rⁿ)/(1 − r) （r ≠ 1）

Alternatively, Sₙ = u₁(rⁿ − 1)/(r − 1) is equivalent. Pick the form that avoids negative denominators in your computation. When adding terms manually, use the formula as a quick check.

等价形式 Sₙ = u₁(rⁿ − 1)/(r − 1) 也可使用。选择能避免分母为负的形式进行计算。手动加和时，用公式快速验证。

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6. Sum to Infinity | 无限项求和

If the absolute value of the common ratio is less than 1 (|r| < 1), the terms get smaller and smaller, and the infinite geometric series converges to a finite sum. This sum to infinity, S∞, appears frequently in IB questions involving recurring decimals, bouncing balls, and financial models.

如果公比的绝对值小于 1（|r| < 1），项会越来越小，无穷等比级数收敛到一个有限和。这个无限项和 S∞ 经常出现在 IB 考题中，涉及循环小数、反弹球和金融模型。

S∞ = u₁ / (1 − r), provided |r| < 1 | S∞ = u₁/(1 − r)，条件是 |r| < 1

Remember, if |r| ≥ 1, the sum to infinity does not exist (it diverges). Also, be meticulous with the domain: the formula only applies from n = 1 to infinity, so ensure u₁ is truly the first term of the infinite series.

记住，如果 |r| ≥ 1，无限项和不收敛（发散）。同时，要仔细关注定义域：公式仅适用于从 n=1 到无穷，因此要确保 u₁ 确实是无穷级数的首项。

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7. Applications: Compound Interest | 应用：复利

Compound interest is a classic geometric sequence application. An initial amount P invested at an annual interest rate i (decimal), compounded annually, grows to P(1 + i)ⁿ after n years. Here, the common ratio is (1 + i), and u₁ = P. For compounding k times per year, the multiplier per period is (1 + i/k).

复利是经典的等比数列应用。初始金额 P 以年利率 i（小数）按年复利时，n 年后增至 P(1 + i)ⁿ。此时公比为 (1 + i)，u₁ = P。若每年复利 k 次，每期乘数为 (1 + i/k)。

Exercise 1D.2 may include finding the term representing a future value, or calculating how many years are needed for an investment to double. That often requires solving rⁿ = target multiple using logarithms.

练习 1D.2 可能包括求代表终值的项，或计算投资翻倍所需的年数。这通常需要通过对数求解 rⁿ = 目标倍数。

Doubling time: n = log 2 / log(1 + i) | 翻倍时间：n = log 2 / log(1 + i)

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

Populations often grow geometrically under ideal conditions. If a population grows by a fixed percentage each year, the model is Pₙ = P₀ × (1 + growth rate)ⁿ. Similarly, radioactive decay or depreciation follows a geometric sequence with r between 0 and 1.

在理想条件下，人口通常呈几何增长。如果人口每年按固定百分比增长，模型为 Pₙ = P₀ × (1 + 增长率)ⁿ。类似地，放射性衰变或折旧遵循公比介于 0 和 1 之间的等比数列。

You will be asked to find the year when a population exceeds a certain threshold, or to calculate the rate given initial and final values. Switch fluently between the nth term formula and the sum formula depending on whether the question asks for a specific term or a total accumulated quantity.

题目可能会要求找出人口超过某阈值的年份，或根据初值和终值计算增长率。根据问题是求特定项还是累计总量，灵活切换使用第 n 项公式和求和公式。

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9. Using Your GDC Efficiently | 高效使用图形计算器

Your GDC can handle geometric sequences via the Sequence app or by creating a list with a recursive formula. In the sequence mode, enter u(n) = u₁ × r^(n−1) to tabulate terms quickly. For solving rⁿ = value, use the numeric solver or graph the function to find intersections.

你的 GDC 可以通过序列应用或利用递推公式生成列表来处理等比数列。在序列模式中输入 u(n) = u₁ × r^(n−1)，快速列出各项。对于求解 rⁿ = 某值，使用数值求解器或绘制函数图像找交点。

When solving Sₙ problems, input the sum formula directly and adjust n until the desired sum is reached. Always store exact values of r and u₁ as variables to avoid rounding errors, especially in multi-step problems from Exercise 1D.2.

求解 Sₙ 问题时，直接输入求和公式并调整 n 直至达到目标总和。务必将 r 和 u₁ 的精确值存储为变量，以避免舍入误差，特别是在练习 1D.2 的多步问题中。

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10. Common Mistakes and How to Avoid Them | 常见错误与规避方法

A typical error is confusing the nth term formula with the sum formula. Remember: uₙ is a single term; Sₙ is the sum of multiple terms. Another mistake is using rⁿ instead of rⁿ⁻¹ for the nth term, which throws off the entire sequence index.

常见错误之一是混淆第 n 项公式与求和公式。记住：uₙ 是单项；Sₙ 是多项之和。另一个错误是将第 n 项公式的指数用 rⁿ 替代 rⁿ⁻¹，这会导致整个序列索引错位。

When r is negative, students often mishandle signs in the sum formula. Double-check by manually summing the first few terms. Also, never forget the convergence condition |r| < 1 for S∞; applying the formula otherwise gives a nonsensical answer.

当 r 为负时，学生常在求和公式中弄错符号。手动加和几项来复查。此外，永远不要忘记 S∞ 的收敛条件 |r| < 1；否则套用公式会得到无意义的答案。

Finally, in applied problems, always interpret your final answer in context. A term number n = 5.2 in years implies that the target is reached during the 6th year; round appropriately and explain your reasoning.

最后，在应用题中，始终结合背景解释最终答案。若年份项数 n = 5.2，意味着目标在第 6 年达成；合理取整并阐明推理。

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11. Step-by-Step Exam-Style Question | 考试型题目分步解析

Question: The third term of a geometric sequence is 12 and the sixth term is 96. Find the first term and the common ratio. Hence, find the sum of the first 10 terms and the sum to infinity if it exists.

题目：一个等比数列的第三项为 12，第六项为 96。求首项和公比，并由此求前 10 项之和以及无穷项和（若存在）。

Step 1: Write equations. u₃ = u₁ r² = 12; u₆ = u₁ r⁵ = 96. Step 2: Divide to get r³ = 8, so r = 2. Step 3: Substitute back: u₁ × 2² = 12 => u₁ = 3. Step 4: S₁₀ = 3(2¹⁰ − 1)/(2 − 1) = 3(1024 − 1) = 3069. Step 5: Since |r| = 2 > 1, sum to infinity does not exist.

步骤 1：列方程。u₃ = u₁ r² = 12；u₆ = u₁ r⁵ = 96。步骤 2：相除得 r³ = 8，故 r = 2。步骤 3：代回，u₁ × 2² = 12 → u₁ = 3。步骤 4：S₁₀ = 3(2¹⁰ − 1)/(2 − 1) = 3(1024 − 1) = 3069。步骤 5：因为 |r| = 2 > 1，无穷项和不存在。

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12. Key Formulas Quick Reference | 核心公式速查

Concept 概念	Formula 公式
nth term 第 n 项	uₙ = u₁ × rⁿ⁻¹
Sum of first n terms 前 n 项和	Sₙ = u₁(1 − rⁿ)/(1 − r), r ≠ 1
Sum to infinity 无穷项和	S∞ = u₁/(1 − r), |r| < 1
Compound growth 复利增长	A = P(1 + i)ⁿ

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