Geometric Sequences: General Term & Properties | 等比数列通项公式与性质

📚 Geometric Sequences: General Term & Properties | 等比数列通项公式与性质

A geometric sequence (or geometric progression, GP) is one of the most fundamental concepts in A-Level mathematics. It appears in topics ranging from algebra and calculus to financial mathematics and exponential modelling. Understanding its general term and properties is essential for solving a wide range of exam problems with confidence.

等比数列（Geometric Progression，简称 GP）是 A-Level 数学中最基础的概念之一。它出现在代数、微积分、金融数学和指数建模等多个专题中。掌握其通项公式与核心性质，是自信解决各类考试题目的关键。

1. Definition and Common Ratio | 定义与公比

A geometric sequence is a sequence 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 \( r \) (we write it as r). For example, \( 3, 6, 12, 24, \dots \) is a geometric sequence with first term \( a = 3 \) and common ratio \( r = 2 \).

等比数列是指从第二项起，每一项与前一项的比值都等于同一个非零常数，这个常数称为公比，通常用 r 表示。例如，\( 3, 6, 12, 24, \dots \) 是一个首项 \( a = 3 \)、公比 \( r = 2 \) 的等比数列。

To determine whether a sequence is geometric, check that the ratio of consecutive terms is constant:

判断一个数列是否为等比数列，只需验证相邻两项的比值是否恒为常数：

r = uₙ₊₁ / uₙ = u₂ / u₁ = u₃ / u₂ = … (constant)

It is important to note that r cannot be zero. If the common ratio is negative, the terms alternate in sign; for example, \( 5, -10, 20, -40, \dots \) has r = −2.

特别要注意的是，公比 r 不能为零。若公比为负数，数列的符号会交替变化；例如，\( 5, -10, 20, -40, \dots \) 的公比就是 r = −2。

2. The General Term Formula | 通项公式

If the first term of a geometric sequence is a and the common ratio is r, then the n-th term (also called the general term) is given by:

若等比数列的首项为 a，公比为 r，则第 n 项（也称为通项）的公式为：

uₙ = a · rⁿ⁻¹

Here, n is a positive integer (n ≥ 1). The exponent is n − 1 because the first term corresponds to n = 1, giving a · r⁰ = a.

其中 n 为正整数（n ≥ 1）。指数是 n − 1，因为当 n = 1 时，第一项为 a · r⁰ = a。

This formula allows us to find any term directly without listing all preceding terms. For instance, for the sequence \( 2, 6, 18, 54, \dots \), we have a = 2 and r = 3; the 6th term is \( 2 \times 3^{5} = 2 \times 243 = 486 \).

利用通项公式，我们可以直接求出任意一项，而无需逐一列出前面的项。例如，对于数列 \( 2, 6, 18, 54, \dots \)，a = 2，r = 3，那么第 6 项为 \( 2 \times 3^{5} = 2 \times 243 = 486 \)。

3. Derivation of the General Term | 通项公式的推导

The general term formula can be derived by observing the pattern of successive multiplication:

通项公式可以通过观察逐项相乘的规律来推导：

u₁ = a
u₂ = a · r
u₃ = a · r²
u₄ = a · r³

From this pattern, we can conclude that the exponent of r is always one less than the term index. Hence, uₙ = a · rⁿ⁻¹. This is a standard result that you should be able to state and apply rapidly in exams.

从以上规律可以看出，r 的指数始终比项数序号少 1。因此，uₙ = a · rⁿ⁻¹。这是一个标准结论，考试中应能迅速写出并熟练应用。

An alternative derivation uses the recurrence relation uₙ₊₁ = r · uₙ. By repeatedly applying this relation from u₁ to uₙ, we multiply by r exactly (n − 1) times, giving the same formula. This recurrence view also connects to later work on iterative processes and exponential functions.

另一种推导方法是利用递推关系 uₙ₊₁ = r · uₙ。从 u₁ 连续应用此关系到 uₙ，相当于乘了 (n − 1) 次 r，得到同样的公式。这种递推视角也为我们后续学习迭代过程和指数函数打下基础。

4. Geometric Mean | 等比中项

If three consecutive terms of a GP are x, y and z, then y is the geometric mean of x and z, satisfying:

若等比数列中连续三项为 x、y、z，则 y 是 x 与 z 的等比中项，满足：

y² = x · z, i.e. y = ±√(x · z)

For positive x and z, the positive square root is usually taken, but the negative value is also valid if the product is positive. For example, if 2 and 8 are two terms, the geometric mean is ±4; the sequence could be \( 2, 4, 8 \) or \( 2, −4, 8 \).

当 x 和 z 均为正数时，通常取正平方根，但若乘积为正，负值同样成立。例如，若两项为 2 和 8，它们的等比中项为 ±4；数列可以是 \( 2, 4, 8 \)，也可以是 \( 2, −4, 8 \)。

This property is frequently tested in problems about inserting one or more terms between two given numbers to form a GP. The common ratio can be found by solving rⁿ⁻¹ = (last term) / (first term).

这一性质常用于在给定两个数之间插入一项或多项以构成等比数列的问题。公比可通过求解 rⁿ⁻¹ = (末项) / (首项) 来得到。

5. Exponential Form of the General Term | 通项的指数形式

Rewriting the general term uₙ = a · rⁿ⁻¹ as uₙ = (a/r) · rⁿ reveals that every geometric sequence is an exponential function of n. This observation is powerful: if r > 1, the terms grow without bound; if 0 < r < 1, the terms decay towards zero; if r < 0, the terms oscillate in sign.

将通项公式改写为 uₙ = (a/r) · rⁿ，可以看出每个等比数列都是关于 n 的指数函数。这一观察非常有力：当 r > 1 时，项无限增大；当 0 < r < 1 时，项递减趋近于零；当 r < 0 时，项的符号正负交替。

This exponential interpretation helps when comparing GPs with arithmetic sequences (AP). In an AP, the difference between consecutive terms is constant, producing linear growth; in a GP, the ratio is constant, producing exponential growth or decay. Recognising which model applies is a key skill in exam questions involving population growth, radioactive decay, or compound interest.

这种指数型解释有助于对比等比数列与等差数列（AP）。在等差数列中，相邻两项之差恒定，呈线性增长；在等比数列中，相邻两项之比恒定，呈指数增长或衰减。在涉及人口增长、放射性衰变或复利等考试题中，识别应使用哪种模型是关键技能。

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

The sum of the first n terms of a geometric sequence, denoted Sₙ, is given by two equivalent forms:

等比数列前 n 项和记为 Sₙ，其公式有两种等价形式：

Sₙ = a(1 − rⁿ) / (1 − r) = a(rⁿ − 1) / (r − 1), for r ≠ 1

When r = 1, the sequence is constant and Sₙ = n · a. The choice between the two forms often depends on whether r < 1 (use the first form to avoid negative denominators) or r > 1 (either form works).

当 r = 1 时，数列为常数数列，Sₙ = n · a。两种形式的选择通常取决于 r 的大小：当 r < 1 时，用第一种形式可避免分母为负；当 r > 1 时，两种形式均可。

To derive the formula, write Sₙ = a + ar + ar² + … + arⁿ⁻¹, then multiply by r and subtract: rSₙ − Sₙ = a(rⁿ − 1), which leads directly to the result. This derivation is a classic exam question in its own right.

推导该公式时，先写出 Sₙ = a + ar + ar² + … + arⁿ⁻¹，再将其乘以 r 后与原式相减：rSₙ − Sₙ = a(rⁿ − 1)，直接得到结果。这一推导过程本身就是经典考题。

7. Sum to Infinity | 无穷等比数列的和

If the common ratio satisfies |r| < 1, the terms become progressively smaller, and the sum of the infinite geometric series converges to a finite value:

当公比满足 |r| < 1 时，每一项越来越小，无穷等比级数收敛于有限值：

S∞ = a / (1 − r), for |r| < 1

This is obtained by letting n → ∞ in the formula for Sₙ, noting that rⁿ → 0 when |r| < 1. If |r| ≥ 1, the series diverges (unless a = 0), and the sum to infinity does not exist.

该公式是在 Sₙ 中令 n → ∞ 得到的，因为当 |r| < 1 时，rⁿ → 0。若 |r| ≥ 1，则级数发散（除非 a = 0），此时无穷和不存在。

For example, the series \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots \) has a = 1 and r = 1/2, so S∞ = 1 / (1 − 1/2) = 2. This result is also the foundation for converting recurring decimals into fractions, a topic often tested in Pure Mathematics papers.

例如，级数 \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots \) 中，a = 1，r = 1/2，因此 S∞ = 1 / (1 − 1/2) = 2。这一结论也是将循环小数化为分数的基础，是纯数学试卷中的常见考点。

8. Key Properties of Geometric Sequences | 等比数列的重要性质

Several properties are frequently used to simplify problems. First, if we multiply every term of a GP by a constant k, the new sequence is still a GP with the same common ratio r. Second, if we take the reciprocal of each term, the resulting sequence is a GP with common ratio 1/r.

等比数列有几条常用性质可用于简化问题。其一，若将等比数列的每一项乘以常数 k，得到的新数列仍是等比数列，且公比仍为 r。其二，若对每一项取倒数，所得新数列仍是等比数列，公比为 1/r。

Third, if two geometric sequences have common ratios r₁ and r₂, then their product (term-by-term) is a GP with common ratio r₁ · r₂. Similarly, the quotient of two GPs (term-by-term) is a GP with common ratio r₁ / r₂.

其三，若两个等比数列的公比分别为 r₁ 和 r₂，则它们逐项相乘所得的新数列仍是等比数列，公比为 r₁ · r₂。同理，两个等比数列逐项相除所得的新数列仍是等比数列，公比为 r₁ / r₂。

Fourth, the terms at equally spaced positions in a GP also form a GP. For example, u₁, u₃, u₅, … form a GP with common ratio r². This property is useful in problems involving odd or even indexed terms.

其四，等比数列中等间隔位置上的项仍然构成等比数列。例如，u₁、u₃、u₅、… 构成公比为 r² 的等比数列。该性质在涉及奇数项或偶数项的问题中非常有用。

9. Problem-Solving Techniques | 解题策略与技巧

When solving GP problems, the most reliable approach is to identify the two parameters a and r from the given conditions. If two independent pieces of information are provided, you can set up two equations in a and r and solve them simultaneously.

解决等比数列问题最可靠的方法是先从题目条件中确定两个参数 a 和 r。若题目给出两个独立条件，则可以建立关于 a 和 r 的两个方程并联立求解。

In many cases, dividing one equation by another eliminates a and leaves an equation solely in r. For example, if u₅ = 32 and u₈ = 256, then u₈ / u₅ = r³ = 8, so r = 2, and a = u₅ / r⁴ = 32 / 16 = 2.

在许多情况下，两式相除可以消去 a，得到只含 r 的方程。例如，若 u₅ = 32，u₈ = 256，则 u₈ / u₅ = r³ = 8，故 r = 2，进而 a = u₅ / r⁴ = 32 / 16 = 2。

When asked to find the number of terms in a finite GP, take logarithms of both sides of the general term equation. For instance, solving 2 × 3ⁿ⁻¹ = 486 gives n − 1 = log₃(243) = 5, so n = 6. Remember that n must be a positive integer.

当题目要求确定有限等比数列的项数时，可对通项方程两边取对数。例如，解 2 × 3ⁿ⁻¹ = 486，可得 n − 1 = log₃(243) = 5，因此 n = 6。注意 n 必须为正整数。

10. Common Pitfalls and Exam Traps | 常见易错点与考试陷阱

One common mistake is confusing the formula for the n-th term with the sum formula. The general term is a · rⁿ⁻¹, while the sum of n terms is a(1 − rⁿ)/(1 − r). Always check the exponent: the term formula has exponent n − 1; the sum formula has exponent n.

常见错误之一是混淆通项公式与求和公式。通项为 a · rⁿ⁻¹，而前 n 项和为 a(1 − rⁿ)/(1 − r)。务必检查指数：通项公式中指数为 n − 1，求和公式中指数为 n。

Another pitfall is forgetting that r = 1 is a special case for the sum formula; using the standard formula with r = 1 causes division by zero. Additionally, when using the sum to infinity, always verify the condition |r| < 1 before applying S∞ = a / (1 − r).

另一个易错点是忘记求和公式中 r = 1 的特殊情况；若直接套用标准公式会导致分母为零。此外，使用无穷求和公式前，务必验证条件 |r| < 1 是否成立。

Finally, be careful with negative common ratios and fractional values of r. Substituting correctly into the formulas with proper brackets is essential to avoid sign errors. When in doubt, test your result with the first few terms by direct multiplication.

最后，要小心负公比和分数形式的 r。代入公式时务必正确使用括号，以避免符号错误。若不确定结果，可用前几项直接相乘来验证。

11. Applications in Real-World Contexts | 实际应用

Geometric sequences appear naturally in many real-world situations. Compound interest is a classic example: if a principal P is invested at an annual interest rate of 6% compounded annually, the amount after n years is P × 1.06ⁿ, which is the (n + 1)-th term of a geometric sequence with first term P and common ratio 1.06.

等比数列在现实生活中有大量自然应用。复利是经典例子：若本金 P 按年利率 6% 复利计息，则 n 年后的金额为 P × 1.06ⁿ，这正是首项为 P、公比为 1.06 的等比数列的第 (n + 1) 项。

Population growth, where a population increases by a fixed percentage each year, follows the same model. Similarly, radioactive decay can be modelled by a geometric sequence with a common ratio between 0 and 1, leading naturally to the concept of half-life.

人口增长——每年按固定百分比增长——遵循同样的模型。类似地，放射性衰变也可以用公比介于 0 和 1 之间的等比数列来建模，并自然引出半衰期的概念。

Present value calculations in finance also rely on geometric series. The present value of a perpetuity, for instance, is computed using the sum to infinity formula, connecting abstract mathematical results directly to practical financial decisions.

金融中的现值计算同样依赖等比级数。例如，永续年金的现值正是通过无穷求和公式计算得到的，这使抽象的数学结论直接服务于实际的财务决策。

12. Summary and Final Advice | 总结与备考建议

The general term uₙ = a · rⁿ⁻¹ and the sum formulas Sₙ = a(1 − rⁿ)/(1 − r) and S∞ = a/(1 − r) for |r| < 1 form the core of the geometric sequence topic. Master these formulas, understand their derivations, and practise applying them to diverse problem types.

通项公式 uₙ = a · rⁿ⁻¹、前 n 项和公式 Sₙ = a(1 − rⁿ)/(1 − r) 以及条件 |r| < 1 下的无穷和公式 S∞ = a/(1 − r)，构成了等比数列专题的核心。掌握这些公式、理解其推导过程，并练习应用于不同类型的题目。

Remember to check whether a sequence is actually geometric, verify the value of r, and be cautious with special cases such as r = 1. With consistent practice, you can confidently handle any GP-related question in your A-Level examination.

请记住：先判断数列是否为等比数列，验证公比 r 的值，并警惕 r = 1 等特殊情况。通过持续练习，你定能自信应对 A-Level 考试中所有与等比数列相关的题目。

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Geometric Sequences: General Term & Properties | 等比数列通项公式与性质