📚 2A Arithmetic Sequences and Series | 2A 等差数列与算术级数
Welcome to our comprehensive guide on arithmetic sequences and series, a fundamental topic in IB Mathematics. Understanding this concept will help you solve problems involving patterns, sums, and real-life applications like savings plans and loan repayments.
欢迎阅读这份关于等差数列与算术级数的全面指南，这是 IB 数学中的一个基础主题。掌握这一概念将帮助你解决涉及模式、求和以及储蓄计划、贷款偿还等实际应用的问题。
1. Introduction to Sequences | 序列简介
A sequence is an ordered list of numbers, where each number is called a term. Sequences can be finite or infinite. For example, 3, 7, 11, 15, 19 is a finite sequence with five terms.
序列是按顺序排列的一列数，其中的每一个数称为一项。序列可以是有限的，也可以是无限的。例如，3, 7, 11, 15, 19 是一个含有五项的有穷序列。
When we talk about a series, we refer to the sum of the terms of a sequence. So, the series corresponding to the above sequence is 3 + 7 + 11 + 15 + 19.
当我们提到级数时，指的是序列各项的和。因此，上述序列对应的级数为 3 + 7 + 11 + 15 + 19。
2. Definition of an Arithmetic Sequence | 等差数列的定义
An arithmetic sequence is a sequence in which the difference between consecutive terms is constant. This constant difference is called the common difference, usually denoted by d.
等差数列是指相邻两项的差恒为常数的序列。这个常数差称为公差，通常用字母 d 表示。
For instance, in the sequence 2, 5, 8, 11, 14, … . If d > 0, the sequence increases; if d < 0, it decreases; if d = 0, all terms are equal.
例如，在序列 2, 5, 8, 11, 14, … 中，公差 d = 3，因为 5 − 2 = 3，8 − 5 = 3，等等。如果 d > 0，序列递增；如果 d < 0，序列递减；如果 d = 0，所有项都相等。
3. Common Difference and the nth Term Formula | 公差与通项公式
n − 1)d, where u₁ is the first term, n is the term number, and d is the common difference.
等差数列的第 n 项（通项）公式为：uₙ = u₁ + (n − 1)d，其中 u₁ 是首项，n 是项数，d 是公差。
t listing all previous terms. For example, to find the 20th term of the sequence 5, 9, 13, … with u₁ = 5 and d = 4, substitute: u₂₀ = 5 + (20 − 1) × 4 = 5 + 76 = 81.
该公式使你能直接求出任意一项，而无需列出前面的所有项。例如，要找出序列 5, 9, 13, … 的第 20 项，其中 u₁ = 5，d = 4，代入得：u₂₀ = 5 + (20 − 1) × 4 = 5 + 76 = 81。
is particularly useful in solving problems where not all information is given explicitly.
你也可以在已知两项时求出公差。如果 uₚ = a 且 u₀ = b，则 d = (b − a) / (q − p)。这在解决某些信息未明确给出的问题时尤为有用。
4. Finding the nth Term: Examples | 求第n项：示例
+ 2d = 12, u₇ = u₁ + 6d = 28. Subtracting gives 4d = 16, so d = 4. Then u₁ = 12 − 2×4 = 4.
示例 1：等差数列的第 3 项为 12，第 7 项为 28。求首项和公差。解：u₃ = u₁ + 2d = 12，u₇ = u₁ + 6d = 28。相减得 4d = 16，故
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