亚学序列

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The following picture represents a pattern. The pattern is an arithmetic . Do you know how to figure out a pattern? Can you graph the pattern?
::以下图片代表一个图案。 图案是一个算术 。 您知道如何解析图案吗 ? 您可以绘制图案吗 ?

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In this concept, you will learn to recognize, extend and graph arithmetic sequences.
::在此概念中,您将学会识别、扩展和图表算术序列。

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Arithmetic Sequence
::亚学序列

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Consider the following images.
::考虑以下图像 。

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You probably saw a pattern right away. If there were another set of boxes, you could probably guess at how many there would be.
::你可能马上看到了一个图案。如果还有另外一套盒子,你可能会猜到有多少个。

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If you saw this same pattern in terms of number of rectangles, it would look like this:
::如果你在矩形数量上看到同样的模式, 它会看起来是这样的:

2, 4, 6, 8, 10

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This set of numbers is called a sequence ; it is a series of numbers that follow a pattern.
::这组数字被称为序列; 是一个模式下的一系列数字。

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If there was another set of boxes, you’d probably guess there would be 12. If you added another number to the sequence, you would write 12. There was a difference of 2 between each two numbers, or terms , in the sequence.
::如果有另一组框, 你可能会猜到会有12个。 如果你在序列中加上另一个数字, 你会写12个。 在序列中,每两个数字或术语相差2个。

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When you have a sequence with a fixed number between each of the terms (a common difference), you call this sequence an arithmetic sequence .
::当您在每一术语( 常见差异) 之间有一个固定编号的序列时, 您将该序列称为算术序列 。

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Let’s look at an example.
::让我们举个例子。

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What is the common difference between each of the terms in the following sequence?
::以下顺序中每个术语之间的共同区别是什么?

7, 12, 17, 22, 27

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First, subtract term 1 from term 2.
::首先,从第2学期中减去第1学期。

$\begin{align*}12-7 = 5\end{align*}$

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Next, subtract term 2 from term 3.
::接下来,从第3学期中减去第2学期。

$\begin{align*}17-12=5\end{align*}$

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Then, continue this subtraction for the rest of the sequence. The table below shows the difference between successive terms in the sequence.
::然后,对顺序的其余部分继续此减法。下表显示顺序中顺序顺序顺序的差别。

Terms	Difference
7
12	$\begin{align*}12-7 = 5\end{align*}$
17	$\begin{align*}17-12=5\end{align*}$
22	$\begin{align*}22-17 = 5\end{align*}$
27	$\begin{align*}27-22=5\end{align*}$

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The answer is 5.
::答案是5。

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The common difference for this sequence is 5.
::这个序列的共同区别是5。

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This is an arithmetic sequence.
::这是一个算术序列。

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Finding the difference between two terms in a sequence is one way to look at sequences. You have used tables of values for several types of equations and you have used those tables of values to create graphs. Graphs are helpful because they are visual representations of the same numbers. When values rise, you can see them rise on a graph. Let’s use the same ideas, then, to graph arithmetic sequences. Let’s look at an example.
::在序列中查找两个术语之间的差异是查看序列的一种方式。您已经为几种方程式使用了数值表,并且使用这些数值表来创建图表。图表很有帮助,因为它们是相同数字的直观表达。当数值上升时,您可以看到它们在一个图表上上升。那么,让我们用同样的思路来绘制算术序列。让我们举一个例子。

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Graph the sequence $\begin{align*}2, 5, 8, 11, 14, 17, \ldots\end{align*}$
::图2,5,8,8,11,14,17,...

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First convert it into a table of values with independent values being the term number and the dependent values being the actual term.
::首先将它转换成一个数值表,独立值为术语数,依附值为实际值。

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Next, use this table to create a graph.
::下一位, 使用此表格创建图表 。

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You can see the pattern clearly in the graph. That is one of the wonderful things about graphing arithmetic sequences.
::您可以在图表中清楚地看到图案的图案。 这是关于图形化算术序列的奇妙之处之一。

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In the graph that we created in the example, each term was expressed as a single point. This is called discrete data . Graphs of discrete data have only the exact points shown. You do not connect the dots with a line. This type of data is usually involves things that are counted in whole numbers like people or boxes.
::在我们在示例中创建的图表中,每个术语以单点表示。这称为离散数据。离散数据图只显示准确的点。您不将点与线条连接。这类数据通常涉及像人或框那样以整数计算的东西。

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Depending on what type of situation you are graphing, you may choose to connect the points with a line. The line demonstrates that there are data points between the points that you have graphed. This is called continuous data and usually involves things like temperature or length that can change fractionally.
::根据您正在绘制的图形类型,您可以选择将点与直线连接。该直线显示,在您绘制的两点之间有数据点。这被称为连续数据,通常涉及温度或长度等可以小数改变的东西。

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So, you can graph sequences and classify them as either discrete or continuous data. Yet another possibility is continuing a sequence in either direction by adding terms that follow the same pattern.
::因此,您可以绘制序列图,并将其分类为离散数据或连续数据。另外一种可能性是,通过添加遵循相同模式的术语,在任一方向上继续一个序列。

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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 the dot pattern.
::早些时候,有人给了你一个点图案的问题。

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You need to determine the pattern for the image below and then graph it.
::您需要确定下方图像的图案, 然后绘制图案 。

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First, let’s draw a table to show the term number, the number of dots, and the common difference.
::首先,让我们绘制一个表格,显示术语数、点数和共同差异。

Term #	# of dots	Difference
1	3
2	5	$\begin{align*}5-3=2\end{align*}$
3	7	$\begin{align*}7-5=2\end{align*}$
4	9	$\begin{align*}9-7=2\end{align*}$
5	11	$\begin{align*}11-9=2\end{align*}$

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The answer is 2.
::答案是2。

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The common difference for this sequence is 2.
::这个序列的共同区别是 2 。

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This is an arithmetic sequence where you add 2 to get the next term in the sequence.
::这是一个算术序列, 您可以在此添加 2 来获得序列中的下一个学期 。

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Next, graph the term number versus the number of dots.
::下一位, 绘制术语数字与点数的比较 。

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Remember that these are discrete points and you are counting dots.
::记住,这些是分开点,你在数点。

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Example 2
::例2

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What is the common difference in the following sequence?
::以下顺序有什么共同的区别?

$\begin{align*}-15, -13, -11, -9 \ldots\end{align*}$

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Use a table to find the difference between successive terms.
::使用表格查找相继任期之间的差异。

Terms	Difference
-15
-13	$\begin{align*}-13-(-15)= 2\end{align*}$
-11	$\begin{align*}-11-(-13)=2\end{align*}$
-9	$\begin{align*}-9-(-11)=2\end{align*}$

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The answer is 2.
::答案是2。

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The common difference for this sequence is 2.
::这个序列的共同区别是 2 。

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This is an arithmetic sequence where you add 2 to get the next term in the sequence.
::这是一个算术序列, 您可以在此添加 2 来获得序列中的下一个学期 。

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Example 3
::例3

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What is the common difference in the following sequence?
::以下顺序有什么共同的区别?

3, 7, 11, 15

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Use a table to find the difference between successive terms.
::使用表格查找相继任期之间的差异。

Terms	Difference
3
7	$\begin{align*}7-3=4\end{align*}$
11	$\begin{align*}11-7=4\end{align*}$
15	$\begin{align*}15-11=4\end{align*}$

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The answer is 4.
::答案是4。

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The common difference for this sequence is 4.
::这一顺序的共同区别是4。

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This is an arithmetic sequence where you add 4 to get the next term in the sequence.
::这是一个算术序列, 您可以在此添加 4 来获得序列中的下一个学期 。

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Example 4
::例4

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What is the common difference in the following sequence? 
::以下顺序有什么共同的区别?

$\begin{align*}18, 8, -2\end{align*}$

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Use a table to find the difference between successive terms.
::使用表格查找相继任期之间的差异。

Terms	Difference
18
8	$\begin{align*}8-18=10\end{align*}$
-2	$\begin{align*}-2-8=-10\end{align*}$

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The answer is -10.
::答案是 -10

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The common difference for this sequence is -10.
::此序列的常见区别是 -10 。

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This is an arithmetic sequence where you subtract 10 to get the next term in the sequence.
::这是一个算术序列, 您可以从中减去 10 来获得序列中的下一个学期 。

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Example 5
::例5

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What is the common difference in the following sequence?
::以下顺序有什么共同的区别?

81, 86, 91, 96

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Use a table to find the difference between successive terms.
::使用表格查找相继任期之间的差异。

Terms	Difference
81
86	$\begin{align*}86-81=5\end{align*}$
91	$\begin{align*}91-86=5\end{align*}$
96	$\begin{align*}96-91=5\end{align*}$

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The answer is 5.
::答案是5。

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The common difference for this sequence is 5.
::这个序列的共同区别是5。

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This is an arithmetic sequence where you add 5 to get the next term in the sequence.
::这是一个算术序列, 您可以在此添加 5 来获得序列中的下一个学期 。

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Review
::回顾

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Write the common difference for each sequence.
::写下每个序列的共同差异 。

1. -9, -7, -5, -3, -1

2. 5.05, 5.1, 5.15, 5.2, 5.25

3. 3, 6, 10, 15, 21, 28

4. 17, 14, 11, 8, 5, 2

5. 10, 9, 8, 7, 6

6. 3, 5, 7, 9, 11

7. 3, 6, 9, 12

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Solve this problem by using what you know about arithmetic sequences.
::使用算术序列的知识来解决这个问题。

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An ant colony invades the caramels in a candy store. The first day they eat a  $\begin{align*}\frac{1}{4}\end{align*}$ of a caramel, the second day  $\begin{align*}\frac{1}{2}\end{align*}$ of a caramel, the third day $\begin{align*}\frac{3}{4}\end{align*}$ .
::蚂蚁聚居地入侵糖果店的焦糖。 第一天他们吃14焦糖,第二天吃12焦糖,第三天是第34天。

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8. What is the difference between each day?
::8. 每一天有什么区别?

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9. How many do you think they’ll eat on the fourth, fifth, and sixth days?
::9. 你认为四、五、六天他们能吃多少?

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Review (Answers)
::回顾(答复)

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To see the answer key for this book, go to the and click on the Answer Key under the ' ' option.
::要查看本书的答案键, 请在“ ” 选项下点击答案键 。