Geometric Sequences: Definition and Applications | 等比数列的定义与应用

📚 Geometric Sequences: Definition and Applications | 等比数列的定义与应用

A geometric sequence is one of the most fundamental concepts in IB Mathematics, appearing in both Analysis & Approaches (AA) and Applications & Interpretation (AI) syllabi. It forms the backbone of topics ranging from exponential growth to financial mathematics.

等比数列是IB数学中最基础的概念之一，出现在分析与方法（AA）和应用与解读（AI）两门课程大纲中。它构成了从指数增长到金融数学等众多主题的基础。

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

A sequence u₁, u₂, u₃, … is geometric if there exists a constant r such that uₙ₊₁ / uₙ = r for all positive integers n. Equivalently, uₙ₊₁ = r × uₙ, where r ≠ 0. This fixed constant r is called the common ratio.

数列 u₁, u₂, u₃, … 是等比数列，当且仅当存在常数 r，使得对任意正整数 n，都有 uₙ₊₁ / uₙ = r。等价地，uₙ₊₁ = r × uₙ，其中 r ≠ 0。这个固定常数 r 称为公比。

For example, the sequence 2, 6, 18, 54, … is geometric with common ratio r = 3, since each term is obtained by multiplying the previous term by 3. Similarly, 100, 50, 25, 12.5, … is geometric with r = 0.5, because each term is half of the previous one.

例如，数列 2, 6, 18, 54, … 是公比 r = 3 的等比数列，因为每一项都是前一项乘以3得到的。类似地，100, 50, 25, 12.5, … 是公比 r = 0.5 的等比数列，因为每一项都是前一项的一半。

• Each term is found by multiplying the previous term by a fixed non-zero number r.

  每一项都是由前一项乘以固定的非零数 r 得到的。

• The common ratio r can be positive, negative, or fractional.

  公比 r 可以为正数、负数或分数。

• If r = 1, the sequence is constant; if r = −1, it alternates between two values.

  若 r = 1，数列为常数列；若 r = −1，数列在两个值之间交替。

2. The Common Ratio r | 公比 r

The common ratio is the defining feature of a geometric sequence. It can be found by dividing any term by its preceding term, provided no term is zero.

公比是等比数列的判别特征。它可以通过任一项除以其前一项得出，前提是数列中没有零项。

r = uₙ₊₁ / uₙ = u₂ / u₁ = u₃ / u₂

For instance, in the sequence 5, −10, 20, −40, …, the common ratio is r = −10/5 = −2. The negative sign causes the terms to alternate in sign.

例如，在数列 5, −10, 20, −40, … 中，公比为 r = −10/5 = −2。负号使各项正负交替。

3. General Term (nth Term) | 通项公式

The nth term of a geometric sequence with first term u₁ and common ratio r is obtained by multiplying u₁ by the common ratio raised to the power (n − 1).

首项为 u₁、公比为 r 的等比数列的第 n 项，等于 u₁ 乘以公比的 (n − 1) 次幂。

uₙ = u₁ × rⁿ⁻¹

This formula follows from repeated multiplication: u₂ = u₁r, u₃ = u₁r², u₄ = u₁r³, and by induction uₙ = u₁rⁿ⁻¹. The exponent is always one less than the term number.

该公式由连续相乘推导：u₂ = u₁r，u₃ = u₁r²，u₄ = u₁r³，归纳可得 uₙ = u₁rⁿ⁻¹。指数总是比项数少1。

Worked Example: Find the 10th term of the geometric sequence 3, 6, 12, 24, …

例题：求等比数列 3, 6, 12, 24, … 的第10项。

Here u₁ = 3 and r = 2. Substituting into uₙ = u₁rⁿ⁻¹ gives u₁₀ = 3 × 2⁹ = 3 × 512 = 1536.

此处 u₁ = 3，r = 2。代入 uₙ = u₁rⁿ⁻¹，得 u₁₀ = 3 × 2⁹ = 3 × 512 = 1536。

4. Geometric Mean | 等比中项

For three consecutive terms a, b, c of a geometric sequence, the relationship b² = ac always holds. The middle term b is called the geometric mean of a and c.

对于等比数列中连续的三项 a,

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