Recurrence Relations | 递推关系

📚 Recurrence Relations | 递推关系

Recurrence relations provide a powerful way to define a sequence when each new term is generated from one or more previous terms. Instead of giving a direct formula such as uₙ = n², a recurrence relation tells us how to move step by step from a known starting point to the next value. In this article we will cover the notation used in Edexcel A-Level Mathematics, methods for generating terms, techniques for solving first-order and second-order linear recurrence relations, and the important Fibonacci example.

递推关系是一种用前一项或前若干项来生成下一项的数列定义方式。与直接给出通项公式（如 uₙ = n²）不同，递推关系告诉我们如何从已知的起点一步一步得到后面的值。本文将围绕 Edexcel A-Level 数学考纲，讲解递推关系的记号、逐项生成的方法、一阶与二阶线性递推关系的求解技巧，以及重要的斐波那契数列例子。

2. What Is a Recurrence Relation? | 什么是递推关系？

A recurrence relation is an equation that defines a sequence recursively. For example, uₙ₊₁ = 2uₙ + 3 is a first-order recurrence relation because each term depends only on the previous term. A second-order relation such as uₙ₊₂ = 5uₙ₊₁ − 6uₙ needs two previous terms to generate the next one.

递推关系是用递归方式定义数列的方程。例如，uₙ₊₁ = 2uₙ + 3 就是一阶递推关系，因为每一项只依赖前一项；而 uₙ₊₂ = 5uₙ₊₁ − 6uₙ 则是二阶递推关系，它需要前两项才能生成下一项。

On its own, a recurrence relation does not produce a unique sequence. We also need initial conditions. For a first-order relation one initial term is enough, for example u₁ = 1; for a second-order relation we usually need two initial terms, for example u₁ = 1 and u₂ = 5. Together, the recurrence relation and the initial conditions define the whole sequence uniquely.

仅有递推关系并不能得到唯一的数列，我们还需要初始条件。一阶递推通常只需要一个初始项，例如 u₁ = 1；二阶递推通常需要两个初始项，例如 u₁ = 1 与 u₂ = 5。递推关系与初始条件合在一起，才能唯一确定整个数列。

3. Notation and Order | 记号与阶数

In Edexcel questions, a sequence is often written as u₁, u₂, u₃, … and the first term is u₁. Two common forms are shown below.

在 Edexcel 的题目中，数列常写成 u₁, u₂, u₃, …，第一项就是 u₁。下面给出两种常见形式。

First order: uₙ₊₁ = f(uₙ)

Second order: uₙ₊₂ = f(uₙ₊₁, uₙ)

The order of a recurrence relation is the number of previous terms needed to find

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