Geometric Sequences | 几何数列

📚 Geometric Sequences | 几何数列

Geometric sequences appear throughout the Edexcel A Level Mathematics specification, particularly in the Pure Mathematics strand. They describe multiplicative patterns in which each term is obtained by multiplying the previous term by a fixed non-zero constant called the common ratio. This article revises the key definitions, formulas, proof ideas, and typical exam applications.

几何数列在 Edexcel A Level 数学考纲中反复出现，尤其在纯数学部分。它描述的是乘法规律：每一项都由前一项乘以一个固定的非零常数（公比）得到。本文梳理关键定义、公式、证明思路及典型考试应用。

1. Definition and Common Ratio | 定义与公比

A geometric sequence, also called a geometric progression, is a list of numbers such that the ratio between any term and the previous term is constant. If the first term is a and the common ratio is r, the sequence has the form a, ar, ar², ar³, … .

几何数列又称等比数列，是一列数，其中任意一项与前一项的比值恒定。若首项为 a，公比为 r，则数列形式为 a, ar, ar², ar³, …。

The common ratio is found by dividing any term by the term immediately before it: r = uₙ / uₙ₋₁. The value r may be positive, negative, or a fraction, but it cannot be zero because that would make all later terms zero and destroy the multiplicative structure.

公比可由任意一项除以其前一项求得：r = uₙ / uₙ₋₁。公比可以为正数、负数或分数，但不能为零，否则后面所有项都为零，破坏乘法结构。

If r > 1, the sequence grows without bound; if 0 < r < 1, each term gets closer to zero. A negative common ratio creates alternating signs.

如果 r > 1，数列无限增长；如果 0 < r < 1，每一项越来越接近零。公比为负会产生正负交替。

2. The nth Term Formula | 第 n 项公式

For a geometric sequence with first term a and common ratio r, the nth term is given by the formula:

对于首项为 a、公比为 r 的几何数列，第 n 项由以下公式给出：

uₙ = a rⁿ⁻¹

This formula comes from observing that the first term has no factor of r, the second term has one factor of r, the third term has two factors, and so on. Therefore the index on r is always n − 1, not n.

该公式来自：首项不含 r，第二项含一个 r，第三项含两个 r，以此类推。所以 r 的指数总是 n − 1，而不是 n。

In Edexcel questions, you may be required to use this formula forwards or backwards: given a and r, find uₙ; or given two terms, form simultaneous equations to find a and r.

在 Edexcel 考题中，你可能需要正用或反用这个公式：已知 a 和 r

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