11.2 描述 " 模式 " 和 " 写法 " 是一个序列的递递递规则

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  描述模式和写作顺序递归规则

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  In 2013, the days a full moon appeared were in the following (with Jan. 1 being Day 1). Write a recursive formula for the sequence.
  ::2013年,满月出现的日子如下(1月1日为第1天),为序列写一个循环公式。

  9, 38, 67, 96, ...

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  Source:  Moongiant
  ::资料来源:月亮。

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  Recursive Rule
  ::递递递规则

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  A recursive rule for a sequence is a formula which tells us how to progress from one term to the next in a sequence. Generally, the variable $\begin{align*}n\end{align*}$ is used to represent the term number. In other words, $\begin{align*}n\end{align*}$ takes on the values 1 (first term), 2 (second term), 3 (third term), etc. The variable, $\begin{align*}a_n\end{align*}$ represents the $\begin{align*}n^{th}\end{align*}$ term and $\begin{align*}a_{n-1}\end{align*}$ represents the term preceding $\begin{align*}a_n\end{align*}$ .
  ::序列的递归规则是一个公式,它告诉我们如何在序列中从一个术语进到下一个术语。一般而言,变量 n 用于代表术语编号。换句话说, n 代表值 1 (第一个术语)、 2 (第二个术语)、 3 (第三个术语)等。变量、 a 表示 n 术语, an- 1 表示前一个术语。

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  Example sequence: $\begin{align*}4, 7, 11, 16, \ldots, a_{n-1}, a_n\end{align*}$
  ::实例序列:4,7,11,16...,an-1,an

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  In the above sequence, $\begin{align*}a_1=4\end{align*}$ , $\begin{align*}a_2=7\end{align*}$ , $\begin{align*}a_3=11\end{align*}$ and $\begin{align*}a_4=16\end{align*}$ .
  ::在上述序列中, a1=4, a2=7, a3=11, a4=16。

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  Let's describe the pattern and write a recursive rule for the sequence: $\begin{align*}9, 11, 13, 15, \ldots\end{align*}$
  ::让我们描述一下这个模式 并写出一个序列的循环规则 911,13,15...

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  First we need to determine what the pattern is in the sequence. If we subtract each term from the one following it, we see that there is a common difference of 29. We can therefore use $\begin{align*}a_{n-1}\end{align*}$ and $\begin{align*}a_n\end{align*}$ to write a recursive rule as follows: $\begin{align*}a_n=a_{n-1}+29\end{align*}$
  ::首先,我们需要确定序列中的模式。 如果我们从后面的顺序中减去每个术语, 我们就会发现有一个共同的29个差异, 因此我们可以使用 an-1 和 an- 1 和 a 来写一个循环规则如下: a= an- 1+ 29

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  Now, let's write a recursive rule for the following sequences.
  ::现在,让我们为以下的顺序 写一个循环规则。

  1. $\begin{align*}3, 9, 27, 81, \ldots\end{align*}$

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  In this sequence, each term is multiplied by 3 to get the next term. We can write a recursive rule: $\begin{align*}a_n=3a_{n-1}\end{align*}$
  ::在此序列中, 每个术语乘以 3 以获得下一个术语 。 我们可以写入一个循环规则 : a= 3an - 1

  2. $\begin{align*}1, 1, 2, 3, 5, 8, \ldots\end{align*}$

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  This is a special sequence called the Fibonacci sequence. In this sequence each term is the sum of the previous two terms. We can write the recursive rule for this sequence as follows: $\begin{align*}a_n=a_{n-2}+a_{n-1}\end{align*}$ .
  ::这是一个叫做 Fibonacci 序列的特殊序列。 在这个序列中, 每个词都是前两个词的总和 。 我们可以为这个序列写入递归规则如下: a=an-2+an-1 。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked to write a recursive formula for the sequence 9, 38, 67, 96, ...
  ::早些时候,你被要求写一个序列9,38,67,96的循环公式...

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  First we need to determine what pattern the sequence is following. If we subtract each term from the one following it, we find that there is a common difference of 29. We can therefore use $\begin{align*}a_{n-1}\end{align*}$ and $\begin{align*}a_n\end{align*}$ to write a recursive rule as follows: $\begin{align*}a_n=a_{n-1}+29\end{align*}$
  ::首先,我们需要确定顺序遵循的模式。如果我们从后面的顺序中减去每个术语,我们就会发现有一个29个共同的区别。 因此,我们可以使用 an-1 和 a 来写一个循环规则如下: a= an-1+29

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  Write the recursive rules for the following sequences.
  ::写入以下序列的递归规则。

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  Example 2
  ::例2

  $\begin{align*}1, 2, 4, 8, \ldots\end{align*}$

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  In this sequence each term is double the previous term so the recursive rule is: $\begin{align*}a_n=2a_{n-1}\end{align*}$
  ::在这个顺序中,每个术语是前一术语的两倍,因此,递归规则是:a=2an-1。

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  Example 3
  ::例3

  $\begin{align*}1, -2, -5, -8, \ldots\end{align*}$

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  This time three is subtracted each time to get the next term: $\begin{align*}a_n=a_{n-1}-3\end{align*}$ .
  ::这一次每次减3次以获得下一学期:a=an-1-3。

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  Example 4
  ::例4

  $\begin{align*}1, 2, 4, 7, \ldots\end{align*}$

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  This one is a little trickier to express. Try looking at each term as shown below:
  ::这是一个比较狡猾的表达方式。 尝试看看下面的每个词 :

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  $$\begin{align*}a_1 &=1 \\ a_2 &=a_1+1 \\ a_3 &=a_2+2 \\ a_4 &=a_3+3 \\ & \ \ \vdots \\ a_n &=a_{n-1}+(n-1)\end{align*}$$
  ::a1=1a2=a1+1a3=a2+2a4=a3+3 *an=an-1+(n-1)

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  Review
  ::回顾

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  Describe the pattern and write a recursive rule for the following sequences.
  ::描述图案并写入以下序列的递归规则。

  1. $\begin{align*}\frac{1}{4}, -\frac{1}{2}, 1, -2 \ldots\end{align*}$
  2. $\begin{align*}5, 11, 17, 23, \ldots\end{align*}$
  3. $\begin{align*}33, 28, 23, 18, \ldots\end{align*}$
  4. $\begin{align*}1, 4, 16, 64, \ldots\end{align*}$
  5. $\begin{align*}21, 30, 39, 48, \ldots\end{align*}$
  6. $\begin{align*}100, 75, 50, 25, \ldots\end{align*}$
  7. $\begin{align*}243, 162, 108, 72, \ldots\end{align*}$
  8. $\begin{align*}128, 96, 72, 54, \ldots\end{align*}$
  9. $\begin{align*}1, 5, 10, 16, 23, \ldots\end{align*}$
  10. $\begin{align*}0, 2, 2, 4, 6, \ldots\end{align*}$
  11. $\begin{align*}3, 5, 8, 12, \ldots\end{align*}$
  12. $\begin{align*}0, 2, 6, 12, \ldots\end{align*}$
  13. $\begin{align*}4, 9, 14, 19, \ldots\end{align*}$
  14. $\begin{align*}\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5} \ldots\end{align*}$
  15. $\begin{align*}4, 5, 9, 14, 23, \ldots\end{align*}$

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  Review (Answers)
  ::回顾(答复)

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