13.13 摘要:序列和系列

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  摘要:序列和系列

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  In this chapter, we learned about:
  ::在本章中,我们了解到:

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  Identifying a Sequence and Finding the nth Term in a Sequence 
  ::确定一个序列并按顺序查找第n期

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  • A sequence is a list of numbers or terms that follow a particular pattern. This pattern can take on many forms.
    ::序列是遵循特定模式的数字或术语列表。该模式可以多种形式出现。
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  • A recursive sequence is a sequence in which the terms depend on the values of the previous terms.
    ::递归序列是指术语取决于先前术语值的顺序。
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  • An arithmetic sequence is a sequence in which the difference between two consecutive terms is constant. We call this difference the common difference.
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    • The formula for the nth term of an arithmetic sequence is $\begin{array}{r}{a}_n=a_1+(n-1)d,\end{array}$  where $\begin{array}{r}{a}_1\end{array}$  is the 1st term and $\begin{array}{r}d\end{array}$  is the common difference.
      ::算术序列 nth 术语的公式是 an=a1+(n-1)d, 其中 a1 是第一个术语, d 是共同差 。
    
    ::算术序列是一个序列,在其中,两个连续任期之间的差异是不变的。我们把这个差异称为共同的区别。算术序列 nth 术语的公式是 an=a1+(n-1)d, 其中 a1 是第一个术语, d 是共同的区别。
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  • A geometric sequence is a sequence in which the ratio between any two consecutive terms, $\begin{array}{r}\frac{a_n}{a_{n-1}}\end{array}$ , is constant. This constant value is called the common ratio.
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    • The formula for the nth term of a geometric sequence is $\begin{array}{r}{a}_n=a_1{r}^{n-1},\end{array}$  where $\begin{array}{r}{a}_1\end{array}$  is the 1st term and $\begin{array}{r}r\end{array}$  is the common ratio.
      ::几何序列 n 的公式是 an=a1rn-1, 其中 a1 是第一个条件, r 是共同比率。
    
    ::几何序列是一个序列, 任何两个连续条件( anan-1) 之间的比率都是恒定的。 这个恒定值被称为共同比率。 几何序列 nth 的公式是 an=1rn-1, 其中 a1 是第一个条件, r 是共同比率 。

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  Identifying a Series and Finding the Sum of a Series 
  ::确定一个系列并查找一个系列的总数

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  • A series is the sum of the terms in a sequence. We often express a series with summation notation (also called sigma notation), which uses the capital Greek letter $\begin{array}{r}\sum ,\end{array}$  sigma.  
    ::A系列是术语按顺序的总和。我们通常用总和符号(也称“西格玛符号”)表示一系列术语,该符号使用希腊大写字母 ,西格玛。
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  • The sum of the finite number of terms of an arithmetic sequence is $\begin{array}{r}{S}_n=\frac{1}{2}n(a_1+a_n),\end{array}$  where $\begin{array}{r}n\end{array}$   is the number of terms, $\begin{array}{r}{a}_1\end{array}$  is the 1st term, and $\begin{array}{r}{a}_n\end{array}$  is the nth term.
    ::算术序列的限定术语数总和为 Sn=12n(a1+an),其中 n 是术语数, a1 是第一个术语, 和 n。
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  • The sum of a finite number of terms of a geometric sequence is $\begin{array}{r}{S}_n=\frac{a_1(1-r^n)}{1-r},\end{array}$   where $\begin{array}{r}n\end{array}$  is the number of terms, $\begin{array}{r}{a}_1\end{array}$  is the 1st term, and $\begin{array}{r}r\end{array}$  is the common ratio.
    ::几何序列的限定术语数总和是 Sn=a1(1-rn)1-r, 其中n为术语数,a1为第一个术语,r为共同比率。
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  • An infinite series is a series with an infinite number of terms.
    ::无限的系列是一个有无限数术语的系列。
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  • If the sum gets close to—that is, "approaches"—a particular number, it is said to converge. If it grows without bound, it is said to diverge.
    ::如果总和接近——也就是“近似”——一个特定的数字,据说它会趋同,如果它没有捆绑地生长,它就会变异。
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  • Since we cannot always find the sums of these series by adding all the terms, we can analyze their behavior by observing the patterns of their partial sums.
    ::由于我们无法总是通过增加所有术语来找到这些系列的总和,我们可以通过观察其部分金额的模式来分析它们的行为。
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  • Given $\begin{array}{r}\underset{n=1}{\overset{\mathrm{\infty }}{\sum }}{a}_1{r}^{n-1},\end{array}$  with $\begin{array}{r}|r|<1\end{array}$ , the sum $\begin{array}{r}{S}_{\mathrm{\infty }}\end{array}$  of all the terms in the geometric sequence is  $\begin{array}{r}{S}_{\mathrm{\infty }}=\frac{a_1}{1-r}.\end{array}$
    ::以1=1=1=1和1+r+1, 几何序列中所有术语的总和S=S*a11-r。

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  Finding a Binomial Expansion
  ::寻找二进制扩展

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  • Pascal's Triangle can be formed by having each row begin and end with a 1. Each "interior" value in each row is the sum of the two numbers above it.
    ::帕斯卡尔的三角形可以通过每行开头和结尾都有一行来形成。 每行的每个“内值”是上方两个数字的总和。
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  • We can also find these values by finding combinations.
    ::我们还可以通过找到组合来找到这些价值。
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  • If you choose $\begin{array}{r}r\end{array}$  objects from a collection of $\begin{array}{r}n\end{array}$  objects, there are $\begin{array}{r}r!\end{array}$  ways of arranging what you choose. In equation form, the number of combinations is  $\begin{array}{r}{}_n{C}_r=\frac{n!}{(n-r)!r!}.\end{array}$
    ::如果您从 n 对象的集合中选择 r 对象, 则有r 方法来安排您选择的物体。 在方程式形式中, 组合数是 nCr=n! (n-r)!r!
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  • We also denote a combination as $\begin{array}{r}{\displaystyle (\genfrac{}{}{0ex}{}{n}{r})}\end{array}$ .
    ::我们还以(nr)表示组合。
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  • The Binomial Theorem is $\begin{array}{r}(a+b{)}^n={\displaystyle (\genfrac{}{}{0ex}{}{n}{0})}{a}^n{b}^0+{\displaystyle (\genfrac{}{}{0ex}{}{n}{1})}{a}^{n-1}{b}^1+{\displaystyle (\genfrac{}{}{0ex}{}{n}{2})}{a}^{n-2}{b}^2+\dots +{\displaystyle (\genfrac{}{}{0ex}{}{n}{n-1})}{a}^1{b}^{n-1}+{\displaystyle (\genfrac{}{}{0ex}{}{n}{n})}{a}^0{b}^n.\end{array}$
    ::Binomial定理是 (a+b)n=(n0)anb0+(n1)an-1b1+(n2)an-2b2+...+(nn-1)a1bn-1+(nn)a0bn。
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  • The rth term in an expansion is $\begin{array}{r}{\displaystyle (\genfrac{}{}{0ex}{}{n}{r-1})}{a}^{n-(r-1)}{b}^{r-1}.\end{array}$
    ::扩展中的“Rth”一词是(nr-1)an-(r-1)br-1。

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  Looking Back, Looking Forward
  ::回顾,展望未来

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  In this chapter, we've covered sequences and series and the underlying idea of patterns. W e can see patterns i n our day-to-day lives  in annuities, the number of petals on a flower, and the flight of an arrow. 
  ::在本章中, 我们覆盖了序列和序列 以及模式的基本概念。 我们可以看到我们日常生活中的 年金模式,花上花瓣的数量, 以及箭的飞逝。

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  In this book, we have often discussed patterns implicitly—linear, polynomial, rational, radical, exponential, and logarithmic patterns. These patterns surround us in our daily lives. With the completion of this book, you now have the skills to consider them mathematically.   
  ::在这本书中,我们经常讨论隐性模式 — — 线性、多面性、理性、激进、指数性和对数模式。 这些模式在我们日常生活中环绕着我们。 随着这本书的完成,你现在有能力用数学来思考这些模式。

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  Chapter Review
  ::回顾章次审查