49 lines
2.2 KiB
Markdown
49 lines
2.2 KiB
Markdown
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---
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title: Recursive Formulas for Arithmetic Sequences
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localeTitle: 算术序列的递归公式
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---
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## 算术序列的递归公式
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### 什么是算术序列？
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**序列**是数字列表，其中对一个数字执行相同的操作以获得下一个数字。 **算术序列** 具体是指通过加或减一个值构成的序列 - 称为**公共差异** - 来获得下一个术语。在 为了有效地讨论序列，我们使用一个公式，在放入索引列表时构建序列。通常，这些公式给出一个字母的名称，后跟括号中的参数，以及构建序列的表达式在右手侧。
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`a(n) = n + 1`
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以上是算术序列的公式的示例。
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### 例子
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序列|式 --------- | --------- 1,2,3,4，... | a（n）= n + 1 3,8,13,18，... | b（n）= 5n - 2
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### 递归公式
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注意：数学家从1开始计数，因此按惯例， `n=1`是第一个术语。所以我们必须定义第一个术语是什么。然后我们有 找出并包括共同的差异。再看看这些例子，
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序列|公式|递归公式 --------- | --------- | ------------------- 1,2,3,4，... | a（n）= n + 1 | a（n）= a（n-1）+ 1，a（1）= 1 3,8,13,18，... | b（n）= 5n - 2 | b（n）= b（n-1）+ 5，b（1）= 3
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### 找到公式（给出第一项的序列）
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```
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1. Figure out the common difference
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Pick a term in the sequence and subtract the term that comes before it.
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2. Construct the formula
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The formula has the form: `a(n) = a(n-1) + [common difference], a(1) = [first term]`
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```
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### 找到公式（给定没有第一项的序列）
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```
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1. Figure out the common difference
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Pick a term in the sequence and subtract the term that comes before it.
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2. Find the first term
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i. Pick a term in the sequence, call it `k` and call its index `h`
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ii. first term = k - (h-1)*(common difference)
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3. Construct the formula
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The formula has the form: `a(n) = a(n-1) + [common difference], a(1) = [first term]`
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```
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#### 更多信息：
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有关此主题的更多信息，请访问
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* [维基百科](https://en.wikipedia.org/wiki/Arithmetic_progression)
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* [可汗学院](https://www.khanacademy.org/math/algebra/sequences/constructing-arithmetic-sequences/a/writing-recursive-formulas-for-arithmetic-sequences)
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