几何序列

播放段落

On the way home from school on the day of the trip downtown, a bunch of students stopped off at the arcade. It was always fun to talk and get something to eat and play a video game or two. Sam and Henry began to play a favorite game of theirs with aliens.
::在市区旅行当天回家的路上,一群学生停在游乐场,聊天、吃点东西和玩一两场电子游戏总是很有趣。 Sam和Henry开始和外星人玩他们最喜欢的游戏。

播放段落

“That has a lot of math in it,” Sasha commented as Henry had his turn. “How do you figure?” Henry asked.
::Sasha评论说,“这里面有许多数学,”Henry轮到他。 “你怎么知道?” 亨利问道。

播放段落

“It just does,” Sasha said convincingly. “Think about it. In this video game, an alien splits into two aliens who then split into two more aliens every 10 minutes.”
::萨沙令人信服地说 : “ 想想吧。 在这个电子游戏中,一个外国人分裂成两个外国人,然后每10分钟分裂成两个外国人。 ”

播放段落

“Good point. So tell me how many aliens there would be after they split 10 times,” Henry challenged.
::”Henry质疑, “好观点。所以告诉我,在他们拆散10次之后,有多少外星人会出现,” Henry质疑。

播放段落

In this concept, you will learn to recognize, extend and graph geometric sequences.
::在这个概念中,你会学会识别、扩展和图形几何序列。

播放段落

Geometric Series
::几何序列

播放段落

There are several different types of sequences that follow patterns. An arithmetic has a fixed sum or difference between each term .
::遵循模式的序列有几种不同的类型。计算每个术语之间有固定总和或差异。

播放段落

Consider the boxes below and you will see another type of sequence.
::考虑下方的框,你会看到另一种序列。

播放段落

Can you see a pattern? The boxes increase each time. Using numbers, the sequence could be written as 1, 4, 16, 64.
::你可以看到一个模式吗? 框每次增加。 使用数字, 序列可以写为 1, 4, 16, 64。

播放段落

Is this an arithmetic sequence? There is a difference of 3 between the first two terms, 12 between the second and third terms, and 48 between the third and fourth terms. Since there is no common difference, this is not an arithmetic sequence.
::这是算术序列吗?前两个任期之间有3个,第二和第三个任期之间有12个,第三和第四个任期之间有48个。由于没有共同的区别,这不是算术序列。

播放段落

Looking at the sequence, you may notice that each term is multiplied by 4 to get the next term. The fifth term in the sequence can then be found by taking the 4 ^th term and multiplying by 4. Therefore the fifth term would be $\begin{align*}64 \times 4=256\end{align*}$ .
::查看顺序时, 您可能会注意到, 每个术语乘以 4 来获得下一任期 。 然后, 序列中的第五个术语可以通过取第四任期和乘以 4 来找到 。 因此, 第五任期将是 64x4= 256 。

播放段落

This is a geometric sequence ; it’s a sequence in which the terms are found by multiplying by a fixed number called the common ratio . In the situation above, the common ratio is 4.
::这是一个几何序列; 它是一个序列, 通过一个固定数字乘以一个称为共同比率的固定数字来找到术语。 在以上情况下, 共同比率是 4 。

播放段落

Once you know the common ratio, then you can figure out the next step in the pattern.
::一旦知道共同比率,你就可以找出模式的下一步。

播放段落

Let’s look at another example.
::让我们再看看另一个例子。

播放段落

What is the common ratio between each of the terms in the sequence?
::顺序中每个术语之间的共同比率是什么?

5,10, 20, 40, 80

播放段落

Use a table to find the common ratio between successive terms.
::使用一个表格来查找相继任期之间的共同比率。

Terms	Common Ratio
$\begin{align*}5\end{align*}$
$\begin{align*}10\end{align*}$	$\begin{align*}\frac{10}{5}=2\end{align*}$
$\begin{align*}20\end{align*}$	$\begin{align*}\frac{20}{10}=2\end{align*}$
$\begin{align*}40\end{align*}$	$\begin{align*}\frac{40}{20}=2\end{align*}$
$\begin{align*}80\end{align*}$	$\begin{align*}\frac{80}{40}=2\end{align*}$

播放段落

The answer is 2.
::答案是2。

播放段落

The common ratio for this sequence is 2.
::此序列的共同比率为 2 。

播放段落

This is a geometric sequence where you multiply by 2 to get the next term in the sequence.
::这是一个几何序列, 您可以乘以 2 来获得序列中的下一个术语 。

播放段落

Let’s look at another example.
::让我们再看看另一个例子。

播放段落

Consider the following sequence: 8, 24, 72, 216, ... What is the next term in the sequence?
::考虑以下顺序:8、24、72、216、......顺序中的下一个任期是什么?

播放段落

First, determine if the sequence is arithmetic or geometric.
::首先,确定序列是算术还是几何。

播放段落

If it is arithmetic, it will have a common difference between successive terms. Use a table to determine the differences.
::如果是算术,则在相继任期之间会有共同的差别。使用表格来确定差别。

Terms	Difference
$\begin{align*}8\end{align*}$
$\begin{align*}24\end{align*}$	$\begin{align*}24-8=16 \end{align*}$
$\begin{align*}72\end{align*}$	$\begin{align*}72-24=48\end{align*}$
$\begin{align*}216\end{align*}$	$\begin{align*}216-72=144 \end{align*}$

播放段落

There is no common difference so the sequence is not arithmetic. If it is geometric, it will have a common ratio between successive terms. Use a table to determine the ratios.
::没有常见的差别, 所以序列不是算术。 如果它是几何, 它将会在连续的术语之间有一个共同的比率。 使用一个表格来确定比率 。

Terms	Common Ratio
$\begin{align*}8\end{align*}$
$\begin{align*}24\end{align*}$	$\begin{align*}\frac{24}{8}=3\end{align*}$
$\begin{align*}72\end{align*}$	$\begin{align*}\frac{72}{24}=3\end{align*}$
$\begin{align*}216\end{align*}$	$\begin{align*}\frac{216}{72}=3\end{align*}$

播放段落

The sequence is geometric with a common ratio of 3.
::序列是几何,共同比率为3。

播放段落

Next, find the next term.
::下个学期请见下个学期

播放段落

Since the common ratio is 3, the fifth term in the sequence will be $\begin{align*}216 \times 3=648\end{align*}$ .
::由于共同比率为3, 顺序中的第五学期为216×3=648。

播放段落

The answer is 648.
::答案是648

播放段落

The sequence will go  $\begin{align*}8,24,72,216,{\color{red}648}\ldots\end{align*}$ .
::顺序是824,72,216,648...

播放段落

It can be useful to graph geometric sequences. To do this, you would create a table of values and then use a coordinate plane to plot the points.
::绘制几何序列可能有用。 为此, 您将创建一个数值表, 然后使用坐标平面绘制点 。

播放段落

Let’s look at an example.
::让我们举个例子。

播放段落

The amount of memory that computer chips can hold in the same amount of space doubles every year. In 1992, they could hold 1MB. Chart the next 15 years in a table of values and show the amount of memory on the same size chip in 2007.
::计算机芯片每年能保持的内存量与空间的倍数相等。 1992年,他们可以持有1MB。 在2007年,用一个数值表显示未来15年的内存量,并显示同一大小芯片上的内存量。

播放段落

First, create a table showing the 15 year span. The amount of space doubles every year so the common ratio is 2. This means to get the next term in the sequence you multiply the previous term by 2.
::首先,创建一张显示15年间隔的表格。每年空间量翻一番,共同比率为2,这意味着在将上一个任期乘以2的顺序中将下一个任期乘以2。

Year	Memory(MB)
1992	1
1993	2
1994	4
1995	8
1996	16
1997	32
1998	64
1999	128
2000	256
2001	512
2002	1024
2003	2048
2004	4096
2005	8192
2006	16384
2007	32768

播放段落

Next, graph the data from the table. Again, you will have discrete data.
::下一位,从表格中绘制数据图。再次,您将拥有离散数据。

播放段落

Examples
::实例

播放段落

Example 1
::例1

播放段落

Earlier, you were given a problem about  the splitting aliens.
::早些时候,有人给了你一个问题 关于分裂的外星人。

播放段落

In the video game that Sasha and Henry play, an alien splits into two aliens who then split into two more aliens every 10 minutes. Henry wants to know how many aliens there will be after 10 times.
::在萨沙和亨利玩的视频游戏中,一个外星人分裂成两个外星人,他们每10分钟分裂成另外两个外星人。亨利想知道10次之后有多少外国人。

播放段落

You know that the aliens split into two every 10 minutes. Therefore the pattern is $\begin{align*}2,4,8,16,32,\ldots\end{align*}$ .
::你知道外星人每10分钟分裂成两只 所以模式是2,4,8,16,32...

播放段落

Continue this pattern for 10 times (or 100 minutes).
::继续此模式十次( 或100分钟) 。

$\begin{align*}2,4,8,16,32,64,128,256,512,1024\end{align*}$

播放段落

The answer is 1024.
::答案是1024

播放段落

After the alien splits 10 times, there will be 1024 aliens in the game.
::外星分裂10次后 游戏中将有1 024名外星人

播放段落

Example 2
::例2

播放段落

Find the common ratio in the sequence.
::在序列中找到常见比例 。

$\begin{align*}\frac{1}{4},\frac{1}{8},\frac{1}{16}\end{align*}$

播放段落

Use a table to find the common ratio between successive terms.
::使用一个表格来查找相继任期之间的共同比率。

Terms	Common Ratio
$\begin{align*}\frac{1}{4}\end{align*}$
$\begin{align*}\frac{1}{8}\end{align*}$	$\begin{align*}\frac{1}{8} \div \frac{1}{4}= \frac{1}{2}\end{align*}$
$\begin{align*}\frac{1}{16}\end{align*}$	$\begin{align*}\frac{1}{16} \div \frac{1}{8}= \frac{1}{2}\end{align*}$

播放段落

The answer is  $\begin{align*}\frac{1}{2}\end{align*}$ .
::答案是12岁

播放段落

The common ratio for this sequence is $\begin{align*}\frac{1}{2}\end{align*}$ .
::这个序列的共同比率是12。

播放段落

This is a geometric sequence where you multiply by $\begin{align*}\frac{1}{2}\end{align*}$  to get the next term in the sequence.
::这是一个几何序列, 乘以 12 以获得序列中下一个学期 。

播放段落

Example 3
::例3

播放段落

Find the common ratio for the sequence 2, 4, 8, 16.
::找出序列 2 、 4 、 8 、 16 的通用比例。

播放段落

Use a table to find the common ratio between successive terms.
::使用一个表格来查找相继任期之间的共同比率。

Terms	Common Ratio
$\begin{align*}2\end{align*}$
$\begin{align*}4\end{align*}$	$\begin{align*}4 \div 2 =2 \end{align*}$
$\begin{align*}8\end{align*}$	$\begin{align*}8 \div 4 =2\end{align*}$
$\begin{align*}16\end{align*}$	$\begin{align*}16 \div 8 =2 \end{align*}$

播放段落

The answer is 2.
::答案是2。

播放段落

The common ratio for this sequence is 2.
::此序列的共同比率为 2 。

播放段落

This is a geometric sequence where you multiply by 2 to get the next term in the sequence.
::这是一个几何序列, 您可以乘以 2 来获得序列中的下一个术语 。

播放段落

Example 4
::例4

播放段落

Find the common ratio for the sequence 1, 7, 49, 343.
::找出序列1、7、49、343的共同比率。

播放段落

Use a table to find the common ratio between successive terms.
::使用一个表格来查找相继任期之间的共同比率。

Terms	Common Ratio
$\begin{align*}1\end{align*}$
$\begin{align*}7\end{align*}$	$\begin{align*}7 \div 1 =7 \end{align*}$
$\begin{align*}49\end{align*}$	$\begin{align*}49 \div 7 =7 \end{align*}$
$\begin{align*}243\end{align*}$	$\begin{align*}343 \div 49 =7 \end{align*}$

播放段落

The answer is 7.
::答案是7个

播放段落

The common ratio for this sequence is 7.
::这个序列的共同比率是7。

播放段落

This is a geometric sequence where you multiply by 7 to get the next term in the sequence.
::这是一个几何序列, 你乘以 7 来获得序列中的下一个学期 。

播放段落

Example 5
::例5

播放段落

Find the common ratio for the sequence 400, 100, 25.
::找到序列400,100,25的通用比率。

播放段落

Use a table to find the common ratio between successive terms.
::使用一个表格来查找相继任期之间的共同比率。

Terms	Common Ratio
$\begin{align*}400\end{align*}$
$\begin{align*}100\end{align*}$	$\begin{align*}\frac{100}{400}=\frac{1}{4}\end{align*}$
$\begin{align*}25\end{align*}$	$\begin{align*}\frac{25}{100}=\frac{1}{4}\end{align*}$

播放段落

The answer is $\begin{align*}\frac{1}{4}\end{align*}$ .
::答案是14岁

播放段落

The common ratio for this sequence is $\begin{align*}\frac{1}{4}\end{align*}$ .
::这个序列的共同比率是14。

播放段落

This is a geometric sequence where you multiply by  $\begin{align*}\frac{1}{4}\end{align*}$  to get the next term in the sequence.
::这是一个几何序列, 乘以 14 以获得序列中的下一个学期 。

播放段落

Review
::回顾

播放段落

Find the common ratio between each term.
::在每个学期之间找出共同比率。

1.  $\begin{align*}-4,20,-100,500,-2500\end{align*}$

2.  $\begin{align*}60,15,\frac{15}{4},\frac{15}{16}\end{align*}$

3.  $\begin{align*} \frac{1}{8},\frac{1}{4},\frac{1}{2},1,2\end{align*}$

4.  $\begin{align*}3,6,12,24\end{align*}$

5.  $\begin{align*}4,2,1,0.5,0.25\end{align*}$

6.  $\begin{align*}12,24,48\end{align*}$

7.  $\begin{align*}\frac{1}{2},1,2\end{align*}$

8.  $\begin{align*}100,50,25,12.5\end{align*}$

播放段落

Identify the following sequences as an arithmetic sequence, a geometric sequence, or neither. For arithmetic sequences, find the common difference. For geometric sequences, find the common ratio.
::将以下序列识别为算术序列、几何序列或两者兼而有之。对于算术序列,找到共同的差别。对于几何序列,找到共同的比率。

9.  $\begin{align*}1,4,7,10,13\end{align*}$

10.  $\begin{align*}180,60,20,6 \frac{2}{3}\end{align*}$

11.  $\begin{align*}102,94,86,78\end{align*}$

12.  $\begin{align*}19,27,35,43,50\end{align*}$

13.  $\begin{align*}5,-50,500,-5000,50000\end{align*}$

14.  $\begin{align*} \frac{1}{6},\frac{1}{12},\frac{1}{24},\frac{1}{48}\end{align*}$

15.  $\begin{align*}99,33,11\end{align*}$

播放段落

Review (Answers)
::回顾(答复)

播放段落

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 。