11.1 按顺序查找下届任期

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  在序列中查找下一个任期

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  You drop a rubber ball from a height of 48 inches. Each time it bounces, it reaches lower and lower heights. The following sequence shows its height with each successive bounce. How high will the ball bounce on its fifth bounce?
  ::你从48英寸高处扔下一个橡胶球, 每次弹跳时,它会到达低处和低处。以下的顺序显示它的高度,每次连续弹跳。 第五个弹跳时球弹跳会有多高?

  48, 36, 27, 20.25,...

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  Sequence of Numbers
  ::数字序列序列

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  When looking at a sequence of numbers, consider the following possibilities.
  ::在查看数字序列时,考虑以下可能性。

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  • There could be a common difference (the same value is added or subtracted) to progress from each term to the next.
    ::从每个术语到下一个术语的进展可能有共同的差别(增加或减去相同的价值)。

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  Example: $\begin{array}{r}5,8,11,14,\dots \end{array}$ (add 3)
  ::示例:5,8,11,14,...(添加3)

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  • There could be a common ratio (factor by which each term is multiplied) to progress from one term to the next.
    ::从一个任期到下一个任期的进度可以有一个共同比率(每个任期乘以因素)。

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  Example: $\begin{array}{r}9,3,1\end{array}$ , $\begin{array}{r}\frac{1}{3}\dots \end{array}$ $\begin{array}{r}(\end{array}$ multiply by $\begin{array}{r}\frac{1}{3})\end{array}$
  ::示例: 9,3,1,13...(乘以13)

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  • If the terms are fractions, perhaps there is a pattern in the numerator and a different pattern in the denominators.
    ::如果术语是分数,则分子中可能有模式,分数中则有不同模式。

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  Example: $\begin{array}{r}\frac{1}{9},\frac{3}{8},\frac{5}{7},\frac{7}{6},\dots \end{array}$ (numerator (+2), denominator (-1))
  ::示例:19,38,57,76.(数字(+2),分母(-1))

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  • If the terms are growing rapidly, perhaps the difference between the term values is increasing by some constant factor.
    ::如果术语迅速增长,也许术语值之间的差别会以某种不变因素增加。

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  Example: $\begin{array}{r}2,5,9,14,\dots \end{array}$ (add 3, add 4, add 5, ...)
  ::实例:2,5,9,14,...(增加3,增加4,增加5,......)

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  • The terms may represent a particular type of number such as prime numbers, perfect squares, cubes, etc.
    ::术语可以代表特定类型的数字,如质数、完美的方形、立方体等。

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  Example: $\begin{array}{r}2,3,5,7,\dots \end{array}$ (prime numbers)
  ::实例:2,3,5,7,...(原始数字)

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  • Consider whether each term is the result of performing an operation on the two prior terms.
    ::审议每一术语是否是按前两个术语进行业务的结果。

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  Example: $\begin{array}{r}2,5,7,12,19,\dots \end{array}$ (add the previous two terms)
  ::实例:2,5,7,12,19,...(加上前两个条件)

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  • Consider the possibility that the value is connected to the term number:
    ::考虑该值与术语编号相关的可能性:

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  Example: $\begin{array}{r}0,2,6,12,\dots \end{array}$
  ::例: 0,2,6,12,...

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  In this example $\begin{array}{r}(0\times 1)=0,(1\times 2)=2,(2\times 3)=6,(3\times 4)=12,\dots \end{array}$
  ::在此示例中( 0x1) = 0, (1x2) = 2, (2x3) = 6, (3x4) = 12,...

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  This list is not intended to be a comprehensive list of all possible patterns that may be present in a sequence but they are a good place to start when looking for a pattern.
  ::这份清单并非意在全面列出序列中可能存在的所有可能模式,但在寻找一个模式时,它们是一个良好的起点。

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  Let's find the next two terms in the following sequences.
  ::让我们在接下来的顺序中找到接下来的两个术语。

  1. $\begin{array}{r}160,80,40,20,\underset{\_}{},\underset{\_}{}\end{array}$

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  Each term is the result of multiplying the previous term by $\begin{array}{r}\frac{1}{2}\end{array}$ . Therefore, the next terms are:
  ::每个术语是前一术语乘以12的结果。 因此,下一个术语是:

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  $\begin{array}{r}\frac{1}{2}(20)=10\end{array}$ and $\begin{array}{r}\frac{1}{2}(10)=5\end{array}$
  ::12(20)=10和12(10)=5

  2. $\begin{array}{r}0,3,7,12,18,\underset{\_}{},\underset{\_}{}\end{array}$

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  The difference between the first two terms $\begin{array}{r}(3-0)\end{array}$ is 3, the difference between the second and third terms $\begin{array}{r}(7-3)\end{array}$ is 4, the difference between the third and fourth terms $\begin{array}{r}(12-7)\end{array}$ is 5 and the difference between the fourth and fifth terms $\begin{array}{r}(18-12)\end{array}$ is 6. Each time we add one more to get the next term. The next difference will be 7, so $\begin{array}{r}18+7=25\end{array}$ for the sixth term. To get the seventh term, we add 8, so $\begin{array}{r}25+8=33\end{array}$ .
  ::前两个任期(3-0)的差别是3,第二和第三任期(7-3)的差别是4,第三和第四任期(12-7)的差别是5,第四和第五个任期(18-12)的差别是6,每次我们再增加一个任期以获得下一个任期,下一个差额是7,因此第六任期为18+7=25,第七任期为8,因此是25+8=33。

  3. $\begin{array}{r}9,5,4,1,3,\underset{\_}{},\underset{\_}{}\end{array}$

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  This sequence requires that we look at the previous two terms. To get the third term, the second term was subtracted from the first: $\begin{array}{r}9-5=4\end{array}$ . To get the fourth term, the third term is subtracted from the second: $\begin{array}{r}5-4=1\end{array}$ . Similarly: $\begin{array}{r}4-1=3\end{array}$ . Now, to get the next terms, continue the pattern:
  ::这个顺序要求我们查看前两个任期。 为了获得第三个任期, 从第一个任期中减去第二个任期: 9-5=4。 为了获得第四个任期, 从第二个任期中减去第三个任期: 5-4=1。 同样: 4-1=3。 现在,为了获得下一个任期,请继续这个模式:

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  $\begin{array}{r}1-3=-2\end{array}$ and $\begin{array}{r}3-(-2)=5\end{array}$
  ::1-32和3-2=5

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked to find how high the ball will bounce on its  fifth bounce. 
  ::早些时候,有人要求你 找出第5次弹跳的球有多高

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  Each successive term in the sequence is the result of multiplying the previous term by $\begin{array}{r}\frac{3}{4}\end{array}$ . Therefore, the next term, the fifth, is:
  ::顺序中的每个连续任期是前一任期乘以34的结果。 因此,下一个任期,即第五个任期,是:

  $\begin{array}{r}\frac{3}{4}(20.25)=15.1875\end{array}$ .

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  Therefore, the ball reaches a height of 15.1875 inches on its fifth bounce.
  ::因此,球的第五次弹跳高度为15.1875英寸。

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  Find the next two terms in each of the following sequences.
  ::在接下来的顺序中查找下两个词。

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  Example 2
  ::例2

  $\begin{array}{r}-5,-1,3,7,\underset{\_}{},\underset{\_}{}\end{array}$

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  Each term is the previous term plus 4. Therefore, the next two terms are 11 and 15.
  ::每个任期为前一任期加4。 因此,下两个任期为11和15。

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  Example 3
  ::例3

  $\begin{array}{r}\frac{1}{3},\frac{2}{3},\frac{7}{9},\frac{5}{6}\underset{\_}{},\underset{\_}{}\end{array}$

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  The pattern here is somewhat hidden because some of the fractions have been reduced. If we “unreduced” the second and fourth terms we get the sequence: $\begin{array}{r}\frac{1}{3}\end{array}$ , $\begin{array}{r}\frac{4}{6}\end{array}$ , $\begin{array}{r}\frac{7}{9}\end{array}$ , $\begin{array}{r}\frac{10}{12},\underset{\_}{},\underset{\_}{}\end{array}$ . Now the pattern can be observed to be that the numerator and denominator each increase by 3. So the next two terms are $\begin{array}{r}\frac{13}{15}\end{array}$ and $\begin{array}{r}\frac{16}{18}\end{array}$ . Reducing the last term gives us the final answer of $\begin{array}{r}\frac{13}{15}\end{array}$ and $\begin{array}{r}\frac{8}{9}\end{array}$ .
  ::这里的图案被隐藏了, 因为某些分数已经缩小。 如果我们“ 未减少” 第二和第三个条件, 我们就会得到顺序 : 13, 46, 79, 1012_,_。 现在可以观察到, 分子和分母每增加3 。 因此, 接下来的两个条件是 1315 和 1618 。 减少最后一个条件给我们1315 和 89 的最后答案是 1315 和 89 。

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  Example 4
  ::例4

  $\begin{array}{r}1,4,9,16,\underset{\_}{},\underset{\_}{}\end{array}$

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  This sequence is the set of perfect squares or the term number squared. Therefore the $\begin{array}{r}{5}^{th}\end{array}$ and $\begin{array}{r}{6}^{th}\end{array}$ terms will be $\begin{array}{r}{5}^2=25\end{array}$ and $\begin{array}{r}{6}^2=36\end{array}$ .
  ::这个序列是完整的方形或词号的平方形。 因此, 第5和第6个术语将是 52=25 和 62=36 。

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  Review
  ::回顾

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  Find the next three terms in each sequence.
  ::在每个序列中找出接下来的三个术语。

  1. $\begin{array}{r}15,21,27,33,\underset{\_}{},\underset{\_}{},\underset{\_}{}\end{array}$
  2. $\begin{array}{r}-4,12,-36,108,\underset{\_}{},\underset{\_}{},\underset{\_}{}\end{array}$
  3. $\begin{array}{r}51,47,43,39,\underset{\_}{},\underset{\_}{},\underset{\_}{}\end{array}$
  4. $\begin{array}{r}100,10,1,0.1,\underset{\_}{},\underset{\_}{},\underset{\_}{}\end{array}$
  5. $\begin{array}{r}1,2,4,8,\underset{\_}{},\underset{\_}{},\underset{\_}{}\end{array}$
  6. $\begin{array}{r}\frac{7}{2},\frac{5}{3},\frac{3}{4},\frac{1}{5},\underset{\_}{},\underset{\_}{},\underset{\_}{}\end{array}$

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  Find the missing terms in the sequences.
  ::在序列中找到缺失的术语 。

  7. $\begin{array}{r}1,4,\underset{\_}{},16,25,\underset{\_}{}\end{array}$
  8. $\begin{array}{r}\frac{2}{3},\frac{3}{4},\underset{\_}{},\frac{5}{6},\underset{\_}{}\end{array}$
  9. $\begin{array}{r}0,2,\underset{\_}{},9,14,\underset{\_}{}\end{array}$
  10. $\begin{array}{r}1,\underset{\_}{},27,64,125,\underset{\_}{}\end{array}$
  11. $\begin{array}{r}5,\underset{\_}{},11,17,28,\underset{\_}{},73\end{array}$
  12. $\begin{array}{r}3,8,\underset{\_}{},24,\underset{\_}{},48\end{array}$
  13. $\begin{array}{r}1,1,2,\underset{\_}{},5,\underset{\_}{},13\end{array}$
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  14. Do any of the problems above have a constant difference? If so, which ones and what is the constant?
      ::上述任何问题是否始终有区别?如果有,是哪些问题和什么问题?
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  15. Do any of the problems above have a common ratio? If so, which ones and what is the ratio?
      ::上述任何问题是否有共同比率?如果有,是哪些问题,比率是多少?

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  Review (Answers)
  ::回顾(答复)

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  Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。