📚 Geometric Sequence | 等比数列
A geometric sequence is one of the fundamental types of sequences studied in IB Mathematics. It appears in topics ranging from financial mathematics to population models and can be combined with series, logarithms, and convergence tests. This article provides a comprehensive revision guide covering definitions, formulas, applications, proofs, and common pitfalls, all aligned with the IB analysis and approaches syllabus.
等比数列是 IB 数学中学习的基础数列类型之一。它出现在从金融数学到人口模型的各种主题中，并可与级数、对数以及收敛性检验结合起来。本文提供全面的复习指南，涵盖定义、公式、应用、证明以及常见陷阱，完全符合 IB 分析与方法的课程大纲。
1. Definition and Common Ratio | 定义与公比
A geometric sequence is a list of numbers where each term after the first is obtained by multiplying the preceding term by a fixed, non-zero constant called the common ratio, denoted by r. For example, in the sequence 3, 6, 12, 24, ... the common ratio is r = 2 because each term is twice the previous one.
等比数列是这样一列数：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，这个常数叫做公比，记为 r。例如，在数列 3, 6, 12, 24, ... 中，公比 r = 2，因为每一项都是前一项的两倍。
The common ratio can be positive or negative, but it cannot be zero. If r > 1, the terms grow in magnitude. If 0 < r < 1, the terms shrink towards zero. When r < 0, the signs of the terms alternate, creating an oscillating sequence.
公比可以是正数或负数，但不能为零。若 r > 1，各项的绝对值越来越大；若 0 < r < 1，各项越来越接近零；当 r < 0 时，各项正负交替出现，形成一个震荡数列。
2. The n-th Term Formula | 第 n 项公式
For a geometric sequence with first term a₁ and common ratio r, the n-th term is given by the explicit formula:
a₉ = a₁ · rⁿ⁻¹
对于首项为 a₁ 、公比为 r 的等比数列，第 n 项的显式公式为：
a₉ = a₁ · rⁿ⁻¹
This means that you can find any term directly without having to list all previous terms. The exponent (n − 1) indicates that the first term is not multiplied by r ed once, and so on. Always verify the index: if you need the 10th term, substitute n = 10.
这意味着你可以直接求出任意一项，而不需要逐一列出之前的项。指数中的 (n − 1) 说明第一项完全不乘 r，第二项乘一次，依此类推。务必确认指标：如果需要第 10 项，代入 n = 10 即可。
3. Sum of the First n Terms (Finite Geometric Series) | 前 n 项之和（有限等比级数）
The sum of the first n terms of a geometric sequence, called a finite geometric series, is calculated with the following formula when r ≠ 1:
S₉ = a₁(1 − rⁿ) / (1 − r)
当 r ≠ 1 时，等比数列前 n 项的和（即有限等比级数）可用以下公式计算：
S₉ = a₁(1 − rⁿ) / (1 − r)
If the common ratio is exactly 1, every term is the same and the series is simply S₉ = n · a₁. The formula is extremely useful for sums of terms like 5 + 15 + 45 + ... up to a certain number of terms. Remember that the numerator can also be written as a₁(rⁿ − 1) divided by (r − 1), which is algebraically equivalent.
如果公比恰好为 1，每一项都相同，级数简化为 S₉ = n · a₁。上述公式在计算像 5 + 15 + 45 + ... 这样的项和时非常有用。请注意分子也可以写成 a₁(rⁿ − 1) 再除以 (r − 1)，它们在代数上是等价的。
4. Sum to Infinity (Infinite Geometric Series) | 无穷项之和（无限等比级数）
When the absolute value of the common ratio is less than 1 (|r| < 1), the terms of a geometric sequence become smaller and smaller, and the sum of all infinitely many terms approaches a finite limit. This sum to infinity is given by:
S∞ = a₁ / (1 − r)
当公比的绝对值小于 1（即 |r| < 1）时，等比数列的各项会越来越小，无穷多项的和趋近于一个有限极限。这个无穷和公式为：
S∞ = a₁ / (1 − r)
For example, the infinite series 1 + 1/2 + 1/4 + 1/8 + ... has a₁ = 1 and r = 1/2, giving S&# damental in understanding convergence and is often used in repeating decimal problems and fractals.
例如，无限级数 1 + 1/2 + 1/4 + 1/8 + ... 的 a₁ = 1，r = 1/2，因此 S∞ = 1 / (1 − 1/2) = 2。这一概念是理解收敛性的基础，常用于循环小数和分形问题中。
5. Convergence Condition and the Common Ratio | 收敛条件与公比范围
The key condition for an infinite geometric series to converge is that the common ratio must satisfy −1 < r < 1, or equivalently |r| < 1. If r ≤ −1 or r the series either diverges by oscillation or grows without bound, so no finite sum exists.
无限等比级数收敛的关键条件是公比满足 −1 < r < 1，即 |r| < 1。如果 r ≤ −1 或 r ≥ 1，级数要么因震荡而发散，要么无限增大，因此不存在有限的和。
verge. Understanding this condition also helps in IB exam questions that ask for the range of r for which a given series is summable to infinity.
在应用无穷求和公式之前，必须检验这一条件。即使是看起来平白无奇的级数，如 2 + 2 + 2 + ...（r = 1），也不收敛。理解该条件还有助于解答 IB 考试中要求确定给定级数可求无穷和的 r 范围的问题。
6. Geometric Mean | 等比中项
If three positive numbers a, G, b form a geometric sequence, then G is called the geometric mean of a and b. It satisfies G² = ab, and thus G = √ab. The geometric mean is widely used in contexts such as average growth rates and proportional comparisons.
如果三个正数 a, G, b 构成等比数列，那么 G 称为 a 与 b 的等比中项。它满足 G² = ab，因此 G = √ab。等比中项广泛用于平均增长率和比例比较等情境。
geometric sequence with four terms, then solve for the common ratio and the missing terms.
在 IB 题目中，你可能会被要求在两个数之间插入一个或多个等比中项。例如，要在 4 与 108 之间插入两个等比中项，可将 4, __, __, 108 视为一个有四项的等比数列，然后解出公比和缺失的项。
7. Applications: Compound Interest | 应用：复利计算
One of the most common real-life applications of geometric sequences is compound interest. If a principal amount P is invested at an annual interest rate i (as a decimal), the amount after n years is given by A₉ = P(1 + i)ⁿ. Here, the common ratio is (1 + i), and the sequence of yearly balances forms a geometric progression.
等比数列最常见的生活应用之一是复利。如果本金 P 以年利率 i（小数形式）投资，那么 n 年后的金额为 A₉ = P(1 + i)ⁿ。这里的公比是 (1 + i)，历年余额形成了一个等比数列。
For example, if you invest $1000 at 5% annual interest, the amounts after 1, 2, and 3 years are $1050, $1102.50, and $1157.62, each being 1.05 times the previous amount. This geometric growth explains why long-term investing can be so powerful, an insight tested regularly in IB exams.
例如，若以 5% 的年利率投资 $1000，1、2、3 年后的金额分别为 $1050、$1102.50 和 $1157.62，每一笔都是前一笔的 1.05 倍。这种几何增长解释了长期投资的强大效力，这也是 IB 考试中经常考查的认识。
8. Applications: Population Growth and Decay | 应用：人口增长与衰减
Geometric sequences also model population changes when the growth or decay rate is constant over discrete time intervals. If a population of size P₀ grows by a fixed percentage each period, the population after n periods is P₉ = P₀(1 + g)ⁿ, where g is the growth rate per period. Similarly, radioactive decay or depreciation can be modeled with a ratio less than 1.
当离散时间间隔内的增长率或衰减率恒定时，等比数列也可以用于人口变化建模。如果初始种群大小为 P₀，每个周期都按固定百分比增长，那么 n 个周期后的种群为 P₉ = P₀(1 + g)ⁿ，其中 g 是每周期增长率。同样，放射性衰变或折旧可以用公比小于 1 的模型来描述。
For exponential decay, the common ratio is between 0 and 1, for instance, if a substance retains 80% of its mass each year, then r tric series for the total lifetime release, often appearing in IB problems that combine sequences with environmental science contexts.
对于指数衰减，公比介于 0 和 1 之间；例如，若某种物质每年保留原来质量的 80%，则 r = 0.8。这会产生一个无限等比级数，用于计算整个周期内的总释放量，这类问题在 IB 中常将数列与环境科学背景相结合。
9. Proving the Sum Formula for Finite Geometric Series | 有限等比级数求和公式的证明
The standard proof of the finite sum formula relies on a clever subtraction. Write the sum S₉ = a₁ + a₁r + a₁r² + ... + a₁rⁿ⁻¹. Multiply both sides by r: rS₉ = a₁r + a₁r² + ... + a₁rⁿ.
有限求和公式的标准证明依赖于巧妙的相减。写出和式 S₉ = a₁ + a₁r + a₁r² + ... + a₁rⁿ⁻¹。等号两边同乘 r：rS₉ = a₁r + a₁r² + ... + a₁rⁿ。
Subtracting the second equation from the first cancels most terms: S₉ − rS₉ = a₁ − a₁rⁿ. Factor the left-hand side: S₉(1 − r) = a₁(1 − rⁿ). Dividing by (1 − r), provided r ≠ 1, yields the formula. This proof is examinable and demonstrates an elegant use of algebraic cancellation.
用第一式减去第二式可消去大部分项：S₉ − rS₉ = a₁ − a₁rⁿ。左边提取公因式：S₉(1 − r) = a₁(1 − rⁿ)。当 r ≠ 1 时两边除以 (1 −
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