1.2扩展数值模式

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  扩展数值模式

  

  [Figure 1]

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  Nicholas is on his way to his cousin's new house. He found the street and is just looking for the house. His cousin lives at number 1644. Nicholas notices that the numbers of the houses on the right side of the street seem to follow a pattern:
  ::Nicholas正在去他表弟新房子的路上。他找到了街道,正在寻找房子。他的表弟住在1644号。Nicholas注意到,街右侧的住房数字似乎遵循了一种模式:

  $\begin{array}{r}1574,1584,1594,\dots \end{array}$   

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  Number 1574 was the first house Nicholas saw on the right. If he assumes that the pattern of how the houses are numbered will continue, how can Nicholas extend the pattern to figure out how many houses away his cousin's house is?
  ::1574号是尼古拉斯右边第一栋房子。 如果他假设房屋编号模式将继续下去,那么尼古拉斯如何扩大模式,以了解他表弟家外有多少栋房子?

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  In this concept, you will learn how to extend numerical patterns.
  ::在此概念中,您将学习如何扩展数字模式。

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  Numerical Patterns
  ::数值模式

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  A  numerical pattern  is a sequence of numbers that has been created based on a rule called a  pattern rule . Pattern rules can use one or more mathematical operations to describe the relationship between consecutive numbers in the sequence.
  ::数字模式是一种数字序列,它是根据一种称为模式规则的规则创建的。 模式规则可以使用一个或多个数学操作来描述序列中连续数字之间的关系。

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  Knowing the pattern rule will help you to  extend  the pattern. To  extend  the pattern means to use the pattern rule to write the numbers that would come next in the sequence.
  ::了解模式规则会帮助您扩展模式。 要扩展模式方法, 请使用模式规则写出序列中下一个数字 。

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  Here is an example.
  ::举一个例子。

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  Find the next two numbers in the following sequence:  $\begin{array}{r}3,6,9,12,\underset{\_}{},\underset{\_}{}\end{array}$ .
  ::以以下顺序查找下两个数字: 3, 6, 9, 12, _, _。

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  First, figure out the pattern rule. This is an ascending pattern so the rule likely involves addition or multiplication.
  ::首先,找出模式规则。这是一个不断上升的模式,因此规则可能涉及增加或乘法。

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  Next, come up with potential pattern rules. Think: "What could you do to 3 to get 6?"
  ::接下来,提出潜在的模式规则。想想: “你对3能做些什么才能得到6?”

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  • You could add 3
    ::您可以添加 3
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  • You could multiply by 2.
    ::你可以乘以2乘以2。
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  • You could do a combination of two or more operations.
    ::您可以将两个或两个以上的操作组合在一起。

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  Now, check if any of these potential pattern rules work with the rest of the sequence.
  ::现在,检查一下这些可能的模式规则 是否与序列的其余部分有效

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  Consider 6 and 9.
  ::考虑6和9。

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  • If you add 3 to 6 you get 9. So the pattern rule "add 3" seems to work.
    ::如果你加上3到6,你得到9 所以模式规则"添加3"似乎有效。

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  Make sure "add 3" works throughout the whole sequence.
  ::确保"加3"整个过程都有效

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  "Add 3" works for the whole sequence.
  ::"加3"为整个序列工作

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  Now, extend the pattern. Apply the pattern rule of "add 3" to the 12 at the end of the pattern.
  ::现在扩展图案。 应用“ 添加 3 ” 的图案规则到图案结尾的 12 。

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  The next number in the pattern will be 15.
  ::该图案的下一个数字将是15。

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  "Add 3" one more time to get the sixth number in the sequence.
  ::"加3" 再来一次 获得序列中的第六位数

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  The answer is that the extended pattern is  $\begin{array}{r}3,6,9,12,\underset{\_}{15},\underset{\_}{18}\end{array}$ .
  ::答案是,延长模式是3,6,9,12,15,18。

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  Sometimes you will be interested in a particular term in the pattern. In order to figure out a particular term, keep extending the pattern until you've reached the term you are looking for.
  ::有时候你会对模式中的某个术语感兴趣。 为了找出一个特定术语, 继续延长该模式, 直到您达到您正在寻找的术语 。

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  Let's look at an example.
  ::让我们举个例子。

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  What is the seventh number in the sequence:  $\begin{array}{r}1,3,9,27,\dots \end{array}$ ?
  ::序列中的第七位数是多少:1,3,9,27,...?

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  First, figure out the pattern rule. This is an ascending pattern so the rule likely involves addition or multiplication.
  ::首先,找出模式规则。这是一个不断上升的模式,因此规则可能涉及增加或乘法。

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  Next, come up with potential pattern rules. Think: "What could you do to 1 to get 3?"
  ::接下来,提出潜在的模式规则。想想: “你能对1做什么才能得到3?”

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  • You could add 2.
    ::你可以加上2个。
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  • You could multiply by 3.
    ::你可以乘以3乘以3
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  • You could do a combination of two or more operations.
    ::您可以将两个或两个以上的操作组合在一起。

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  Now, check if any of these potential pattern rules work with the rest of the sequence.
  ::现在,检查一下这些可能的模式规则 是否与序列的其余部分有效

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  Consider 1 and 3.
  ::考虑1和3。

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  • If you add 2 to 3 you get 5, not 9. So the pattern rule is not "add 2."
    ::如果你加上2到3,你得到5,而不是9 所以模式规则不是"加2"
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  • If you multiply 3 by 3 you get 9. So the pattern rule "multiply by 3" seems to work.
    ::如果乘以3乘以3乘以3,你就会得到9 所以模式规则"乘以3"似乎有效

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  Make sure "multiply by 3" works throughout the whole sequence.
  ::确保"乘以3" 在整个序列中有效

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  "Multiply by 3" works for the whole sequence.
  ::整个序列都用"3"来进行"3"

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  Finally, you can extend the pattern to find the seventh number. Keep multiplying by 3 until you reach the seventh number in the sequence.
  ::最后, 您可以扩展图案以找到第七位数。 继续乘以 3 直至您在序列中达到第七位数 。

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  The first seven numbers in the sequence are 1, 3, 9, 27, 81, 243, 729.
  ::顺序的前七个数字是 1,3,9,27,81,243,729

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  The answer is that the seventh number in the sequence is 729.
  ::答案是序列中的第七位数是729

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given a problem about Nicholas, who is going to his cousin's house. 
  ::早些时候,你得到一个问题 关于尼古拉斯, 谁要去他表弟的房子。

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  He just passed house number 1574 and noticed that the house numbers seem to follow a pattern:
  ::他刚通过1574号房 发现房号似乎遵循了一种模式

  $\begin{array}{r}1574,1584,1594\end{array}$

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  His cousin lives at number 1644. Nicholas wants to extend the pattern to predict how many houses away his cousin's house is.
  ::Nicholas想扩大这个模式 预测他表弟家外面有多少房子

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  First, figure out the pattern rule. This is an ascending pattern so the rule likely involves multiplication or addition.
  ::首先,找出模式规则。这是一个不断上升的模式,因此规则可能涉及乘法或加法。

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  Next, look at how the numbers are related. The difference between the numbers is a constant 10. This means that the pattern rule is "add 10." Double check that this pattern rule works for the three given numbers in the sequence.
  ::接下来,请看看数字是如何关联的。数字之间的差别是恒定的 10 。 这意味着模式规则是“ 添加 10 ” 。 双重检查此模式规则是否适用于序列中的三个给定数字 。

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  The pattern rule "add 10" works for the three given numbers in the sequence.
  ::模式规则“ 添加 10” 适用于序列中的三个给定数字 。

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  Now, extend the pattern. Apply the pattern rule "add 10" at the end of the pattern until you hit 1644.
  ::现在, 扩展图案。 在图案结尾处应用“ 添加 10 ” 的图案规则, 直到你到达 1644 。

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  Next, write out the extended sequence.
  ::接下来,写出延长的顺序。

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  Notice that 1644 is the eighth number in the sequence.
  ::注意1644是序列中的第八位数

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  The answer is that Nicholas can expect his cousin's house to be the eighth house on the right.
  ::答案是尼古拉斯可以指望 他表弟的房子是右边第八栋房子

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  Example 2
  ::例2

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  Find the next two numbers in the following sequence:  $\begin{array}{r}24,14,9,\underset{\_}{},\underset{\_}{}\end{array}$ .
  ::按以下顺序查找下两个数字:24、14、9、_、_。

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  First, figure out the pattern rule. This is a descending pattern so the rule likely involves division or subtraction.
  ::首先,找出模式规则。这是一个递减模式,因此规则可能涉及分割或减法。

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  Next, look at how the numbers are related. The difference between the numbers decreases as you move through the sequence. To get from 24 to 14 you have to subtract 10, but to get from 14 to 9 you have to subtract 5. This means division is involved. Since there is no whole number you can divide 24 by to get 14, the pattern rule is likely division with either addition or subtraction.
  ::接下来,请看看数字是如何关联的。随着您在序列中移动,数字之间的差别会缩小。要从24到14,您必须减去10,但要从14到9,您必须减去5。这意味着涉及到除法。既然没有全部数字,您可以将24除以14,那么,模式规则可能是除法,增加或减法。

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  Now, consider possible pattern rules that involve division and addition or subtraction. Think: "What could you do to 24 to get 14?"
  ::现在,考虑一下可能涉及分割和增加或减法的模式规则。想想: “你对24能做些什么才能得到14?”

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  • You could divide by 2 and add 2.
    ::您可以除以 2 并添加 2 。
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  • You could divide by 3 and add 6.
    ::你可以除以3再加6
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  • You could do some other combination of two or more operations.
    ::你可以做其他两种或多种操作的组合。

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  Next, look back at the rest of the sequence. Consider 14 and 9.
  ::接下来,看看其余的顺序,考虑14和9。

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  • If you divide 14 by 2 and add 2 you get 9. So the pattern rule "divide by 2 and add 2" works for the three given numbers in the sequence.
    ::如果将14除以2再加2除以14,则将获得9。因此,模式规则“divide 乘以2再加2”对序列中的3个给定数字起作用。

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  Now, extend the pattern. Apply the pattern rule of "divide by 2 and add 2" at the end of the pattern two times.
  ::现在, 扩展图案。 在图案结尾处应用“ 将二除以 2 并添加 2 ” 的图案规则 两次 。

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  The next two numbers in the sequence will be 6.5 and 5.25.
  ::接下来两个序列中的数字将是6.5和5.25。

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  The answer is that the extended pattern is  $\begin{array}{r}24,14,9,\underset{\_}{6.5},\underset{\_}{5.25}\end{array}$ .
  ::答案是,延长的模式是24、14、9、6.5、5.25_。

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  Example 3
  ::例3

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  Find the next two numbers in the following sequence:  $\begin{array}{r}9,17,33,\underset{\_}{},\underset{\_}{}\end{array}$ .
  ::按以下顺序查找下两个数字: 9、17、33、_、_。

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  First, figure out the pattern rule. This is an ascending pattern so the rule likely involves multiplication or addition.
  ::首先,找出模式规则。这是一个不断上升的模式,因此规则可能涉及乘法或加法。

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  Next, look at how the numbers are related. The difference between the numbers increases as you move through the sequence. This means multiplication is involved. Since there is no whole number you can multiply 9 by to get 17, the pattern rule is likely multiplication with either addition or subtraction.
  ::接下来,请看看数字是如何关联的。随着您在序列中移动,数字之间的差别会增加。这意味着要进行乘法。既然没有整数,您可以乘9乘17,那么模式规则可能会随着增减而倍增。

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  Now, consider possible pattern rules that involve multiplication and addition or subtraction. Think: "What could you do to 9 to get 17?"
  ::现在,考虑一下可能涉及乘法和加法或减法的模式规则。想想: “你对9能做些什么才能得到17?”

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  • You could multiply by 2 and subtract 1.
    ::您可以乘以 2 乘以 2 减去 1 。
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  • You could multiply by 3 and subtract 10.
    ::您可以乘以 3 乘以 3 减去 10 。
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  • You could do some other combination of two or more operations.
    ::你可以做其他两种或多种操作的组合。

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  Next, look back at the rest of the sequence. Consider 17 and 33.
  ::接下来,看看其余的顺序,考虑17和33

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  • If you multiply 17 by 2 and subtract 1 you get 33. So the pattern rule multiply by 2 and subtract 1 works for the three given numbers in the sequence.
    ::如果乘以 17 乘以 2 减去 1 , 就会得到 33 。 因此, 模式规则乘以 2 , 减去 1 对序列中的 3 个给定数字起作用 。

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  Now, extend the pattern. Apply the pattern rule of "multiply by 2 and subtract 1" at the end of the pattern two times.
  ::现在, 扩展图案。 在图案结尾处应用“ 乘以 2 乘以 2 和 减 1 ” 的图案规则 两次 。

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  The next two numbers in the pattern will be 65 and 129.
  ::接下来两个数字将分别是65和129。

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  The answer is that the extended pattern is  $\begin{array}{r}9,17,33,\underset{\_}{65},\underset{\_}{129}\end{array}$ .
  ::答案是,延长的模式是 9, 17, 33, 65, 129。

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  Example 4
  ::例4

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  Find the next two numbers in the following sequence:  $\begin{array}{r}3,10,31,\underset{\_}{},\underset{\_}{}\end{array}$ .
  ::以以下顺序查找下两个数字: 3, 10, 31, _, _。

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  First, figure out the pattern rule. This is an ascending pattern so the rule likely involves multiplication or addition.
  ::首先,找出模式规则。这是一个不断上升的模式,因此规则可能涉及乘法或加法。

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  Next, look at how the numbers are related. The difference between the numbers increases as you move through the sequence. This means multiplication is involved. Since there is no whole number you can multiply 3 by to get 10, the pattern rule is likely multiplication with either addition or subtraction.
  ::接下来,请看看数字是如何关联的。随着您在序列中移动,数字之间的差别会增加。这意味着要进行乘法。既然没有整数,您可以乘以3乘以10,则模式规则可能会随着增加或减法而乘以倍增。

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  Now, consider possible pattern rules that involve multiplication and addition or subtraction. Think: "What could you do to 3 to get 10?"
  ::现在,考虑一下可能涉及乘法和加法或减法的模式规则。想想: “你对3能做些什么才能得到10?”

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  • You could multiply by 2 and add 4.
    ::您可以乘以 2 乘以 2 并添加 4 。
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  • You could multiply by 3 and add 1.
    ::您可以乘以 3 并添加 1 。
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  • You could do some other combination of two or more operations.
    ::你可以做其他两种或多种操作的组合。

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  Next, look back at the rest of the sequence. Consider 10 and 31.
  ::接下来,看看剩下的顺序,考虑10和31

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  • If you multiply 10 by 3 and add 1 you get 31. So the pattern rule "multiply by 3 and add 1" works for the three given numbers in the sequence.
    ::如果乘以 10 乘以 3 , 加上 1 , 得到 31 。 因此, 模式规则“ 乘以 3 乘以 3 , 加上 1 ” 对序列中的 3 个给定数字有效 。

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  Now, extend the pattern. Apply the pattern rule "multiply by 3 and add 1" at the end of the pattern two times.
  ::现在, 扩展图案。 在图案结尾处应用模式规则“ 乘以 3 并增加 1 ” 两次 。

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  The next two numbers in the pattern will be 94 and 283.
  ::接下来两个数字将是94和283。

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  The answer is that the extended pattern is  $\begin{array}{r}3,10,31,\underset{\_}{94},\underset{\_}{283}\end{array}$ .
  ::答案是延长模式是 3, 10, 31, 94, 283_。

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  Example 5
  ::例5

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  Find the sixth number in the following sequence:  $\begin{array}{r}4,17,56,\dots \end{array}$
  ::在以下顺序中查找第六个数字: 4, 17, 56,...

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  First, figure out the pattern rule. This is an ascending pattern so the rule likely involves multiplication or addition.
  ::首先,找出模式规则。这是一个不断上升的模式,因此规则可能涉及乘法或加法。

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  Next, look at how the numbers are related. The difference between the numbers increases as you move through the sequence. This means multiplication is involved. Since there is no whole number you can multiply 4 by to get 17, the pattern rule is likely multiplication with either addition or subtraction.
  ::接下来,请看看数字是如何关联的。随着您在序列中移动,数字之间的差别会增加。这意味着要进行乘法。既然没有整数,您可以乘以4乘以17,则模式规则可能随着增减而增加。

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  Now, consider possible pattern rules that involve multiplication and addition or subtraction. Think: "What could you do to 4 to get 17?"
  ::现在,考虑一下可能涉及乘法和加法或减法的模式规则。想想: “你能对4做什么才能得到17?”

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  • You could multiply by 2 and add 9.
    ::您可以乘以 2 乘以 2 并添加 9 。
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  • You could multiply by 3 and add 5.
    ::您可以乘以 3 乘以 3 , 加上 5 。
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  • You could multiply by 4 and add 1.
    ::您可以乘以 4 并添加 1 。
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  • You could do some other combination of two or more operations.
    ::你可以做其他两种或多种操作的组合。

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  Next, look back at the rest of the sequence. Consider 17 and 56.
  ::接下来,看看其余的顺序,考虑17和56

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  • If you multiply 17 by 3 and add 5 you get 56. So the pattern rule "multiply by 3 and add 5" works for the three given numbers in the sequence.
    ::乘以 17 乘以 3 乘以 3 乘以 5 , 乘以 56 。 因此, 模式规则“ 乘以 3 乘以 5 ” 对序列中的三个给定数字有效 。

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  Now, extend the pattern. Apply the pattern rule of "multiply by 3 and add 5" at the end of the pattern three times in order to get to the sixth number in the sequence.
  ::现在, 扩展图案。 在图案结尾处应用“ 乘以 3 ” 的图案规则, 并加上 5 3 次, 以便达到序列中的第六位数 。

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  The extended pattern is 4, 17, 56, 173, 524, 1577.
  ::延长模式为4、17、56、173、524、1577。

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  The answer is that the sixth number in the sequence is 1577.
  ::答案是序列中的第六位数是1577

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  Review
  ::回顾

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  Extend each numerical pattern by filling in the blanks.
  ::通过填充空白来扩展每个数字模式。

  1. $\begin{array}{r}2,3,4,5,\underset{\_}{},\underset{\_}{}\end{array}$
  2. $\begin{array}{r}2,4,6,8,\underset{\_}{},\underset{\_}{}\end{array}$
  3. $\begin{array}{r}2,5,11,23,\underset{\_}{},\underset{\_}{}\end{array}$
  4. $\begin{array}{r}3,6,9,\underset{\_}{},\underset{\_}{}\end{array}$
  5. $\begin{array}{r}64,16,4,\underset{\_}{},\underset{\_}{}\end{array}$
  6. $\begin{array}{r}150,100,50,\underset{\_}{},\underset{\_}{}\end{array}$
  7. $\begin{array}{r}10,20,30,40,\underset{\_}{},\underset{\_}{}\end{array}$
  8. $\begin{array}{r}15,30,45,\underset{\_}{},\underset{\_}{}\end{array}$
  9. $\begin{array}{r}100,112,124,\underset{\_}{},\underset{\_}{}\end{array}$
  10. $\begin{array}{r}4,18,74,\underset{\_}{},\underset{\_}{}\end{array}$
  11. $\begin{array}{r}40,120,360,\underset{\_}{},\underset{\_}{}\end{array}$
  12. $\begin{array}{r}2.5,6,13,\underset{\_}{},\underset{\_}{}\end{array}$
  13. $\begin{array}{r}50,25,12.5,\underset{\_}{},\underset{\_}{}\end{array}$
  14. $\begin{array}{r}3,4.5,6,7.5,9,\underset{\_}{}\end{array}$

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