节: 亚学和几何序列 | 13.3 测量和几何序列

章节大纲

Introduction
::导言

The image below represents a pattern. The pattern is an arithmetic . Can you identify the pattern and write it as a general equation?
::下面的图像代表一个图案。图案是一个算术。您能够识别图案并将其写成一个普通方程式吗 ?

In this concept, you will learn to recognize, extend, and graph arithmetic .
::在这一概念中,你会学会识别、扩展和图表算术。

Arithmetic Sequence
::亚学序列

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

You probably saw a pattern right away. If there were another set of boxes, you could probably guess 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,你也许可以猜到有多少个。

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.

This set of numbers is called a sequence. A sequence is a list of numbers with a common pattern. The common pattern in an arithmetic sequence is that the same number is added to each number to produce the next number.
::此组数字被称为序列。序列是带有共同模式的数字列表。算术序列中的常见模式是, 将相同数字添加到每个数字中, 以产生下一个数字。

are defined by an initial value $\begin{aligned} a_{1} \end{aligned}$ and a common difference $\begin{aligned} d \end{aligned}$ .
::由初始值a1和共同差 d定义。

$\begin{aligned} a_{1} & = a_{1} \\ a_{2} & = a_{1} + d \\ a_{3} & = a_{1} + 2 d \\ a_{4} & = a_{1} + 3 d \\ ⋮ \\ a_{n} & = a_{1} + (n - 1) d \end{aligned}$

::a1=a1a2=a1+da3=a1+2da4=a1+3d=an=a1+(n-1)d

Play, Learn, and Explore with Arithmetic Sequences:
::与自学序列一起玩、学习和探索:

The following video introduces arithmetic sequences:
::以下视频介绍算术序列:

Geometric Sequence
::几何序列

The common pattern in a geometric sequence is that the same number is multiplied or divided to each number to produce the next number.
::几何序列的共同模式是,同一数字乘以或除以每个数字,得出下一个数字。

are defined by an initial value $\begin{aligned} a_{1} \end{aligned}$ and a common ratio $\begin{aligned} r \end{aligned}$ .
::由初始值a1和共同比率r确定。

$\begin{aligned} a_{1} & = a_{1} \\ a_{2} & = a_{1} \cdot r \\ a_{3} & = a_{1} \cdot r^{2} \\ a_{4} & = a_{1} \cdot r^{3} \\ ⋮ \\ a_{n} & = a_{1} \cdot r^{n - 1} \end{aligned}$

::a1=a1a2=a1-ra3=a1_r2a4=a1_r3_aan=a1_rn_1

If a sequence does not have a common ratio or a common difference, it is neither an arithmetic nor a geometric sequence. You should still try to figure out the pattern and come up with a formula that describes it.
::如果序列没有共同比率或共同差异,它既不是算术,也不是几何序列。你仍应尝试找出模式,并拿出一个描述它的公式。

Play, Learn, and Explore with Geometric Sequences:
::与几何序列一起玩、学习和探索:

The following video introduces geometric sequences:
::以下视频介绍几何序列:

Examples
::实例

Example 1
::例1

For each of the following three sequences, determine if it is arithmetic, geometric, or neither:
::对于下列三个序列的每一个序列,确定它是算术、几何还是两者兼有:

1) $\begin{aligned} 0.135, 0.189, 0.243, 0.297, \dots \end{aligned}$

Solution:
::解决方案 :

The sequence is arithmetic because the common difference is 0.054.
::顺序是算术因为共同差数是0.054

2) $\begin{aligned} \frac{2}{9}, \frac{1}{6}, \frac{1}{8}, \dots \end{aligned}$

Solution:
::解决方案 :

The sequence is geometric because the common ratio is $\begin{aligned} \frac{3}{4} \end{aligned}$ .
::顺序是几何,因为共同比率是34。

3) $\begin{aligned} 0.54, 1.08, 3.24, \dots \end{aligned}$

Solution:
::解决方案 :

The sequence is not arithmetic because the differences between consecutive terms are 0.54 and 2.16, which are not common. The sequence is not geometric because the ratios between consecutive terms are 2 and 3, which are not common.
::顺序不是算术,因为连续任期之间的差别是0.54和2.16,这是不常见的。顺序不是几何,因为连续任期之间的比率是2和3,这是不常见的。

Example 2
::例2

For the following sequence, determine the common ratio or difference, the next three terms, and the 2,013th term .
::对于以下顺序,确定共同比率或差异、后三个任期和第2 013个任期。

$\begin{aligned} \frac{2}{3}, \frac{5}{3}, \frac{8}{3}, \frac{11}{3}, \dots \end{aligned}$

Solution:
::解决方案 :

The sequence is arithmetic because the difference is exactly 1 between consecutive terms. The next three terms are $\begin{aligned} \frac{14}{3}, \frac{17}{3}, \frac{20}{3} \end{aligned}$ . An equation for this sequence would be
::顺序是算术,因为相连任期之间的差差是一。接下来的三个任期是143,173,203。这个序列的方程式是:

$\begin{aligned} a_{n} = \frac{2}{3} + (n - 1) \cdot 1. \end{aligned}$

::an=23+(n-1)______________________________________________________________________________________________________

Therefore, the 2,013th term requires 2,012 times the common difference added to the first term.
::因此,第2,013个任期需要2,012倍于第一个任期所增加的共同差额。

$\begin{aligned} a_{2, 013} = \frac{2}{3} + 2, 012 \cdot 1 = \frac{2}{3} + \frac{6, 036}{3} = \frac{6, 038}{3} \end{aligned}$

::a2,013=23+2,0121=23+6,0363=6,0383

Example 3
::例3

For the following sequence, determine the common ratio or difference and the next three terms:
::对于以下顺序,确定共同比率或差异,以及下三个术语:

$\begin{aligned} \frac{2}{3}, \frac{4}{9}, \frac{6}{27}, \frac{8}{81}, \frac{10}{243}, \dots \end{aligned}$

Solution:
::解决方案 :

This sequence is neither arithmetic nor geometric. The differences between the first few terms are $\begin{aligned} - \frac{2}{9}, - \frac{2}{9}, - \frac{10}{81}, - \frac{14}{243} \end{aligned}$ . While there was a common difference at first, this difference did not hold through the sequence. Be sure to always check the sequence in multiple places to make sure the common difference holds up throughout.
::这个序列既不是算术,也不是几何。前几个术语之间的差别是-29,-29,-1081,-14243。虽然最初有一个共同的差异, 但这一差异并没有贯穿于序列中。一定要检查多个地方的序列, 以确保共同差异始终维持不变。

The sequence is also not geometric because the ratios between the first few terms are $\begin{aligned} \frac{2}{3}, \frac{1}{2}, \frac{4}{9} \end{aligned}$ . These ratios are not common.
::顺序也不是几何,因为前几个条件之间的比率是23,12,49。这些比率并不常见。

Even though you cannot get a common ratio or a common difference, it is still possible to produce the next three terms in the sequence by noticing that the numerator is an arithmetic sequence with starting term of 2 and a common difference of 2. The denominators are a geometric sequence with an initial term of 3 and a common ratio of 3. The next three terms are
::即使你无法获得共同比率或共同差异,但仍有可能在顺序中得出下三个术语的顺序,其方法是注意到分子是一个算算序列,起始期为2,共同差为2。分母是一个几何序列,初始期为3,共同比率为3。

$\begin{aligned} \frac{12}{3^{6}}, \frac{14}{3^{7}}, \frac{16}{3^{8}} . \end{aligned}$

Example 4
::例4

Recall the problem from the Introduction: Can you identify the pattern and write it as a general equation?
::回顾导言中的问题:您能否辨别模式并将其写成一般方程式?

Solution:
::解决方案 :

Each time, 2 dots are added to the previous image. Thus, this sequence is an arithmetic sequence with a common difference of 2. The equation for this sequence is then
$\begin{aligned} a_{n} = 3 + 2 (n - 1) . \end{aligned}$
Example 5
::每次向上一个图像添加两个点。因此, 此序列是一个算术序列, 共差为 2 。此序列的方程式是 =3+2( n- 1) 。例如 5

What is the 10th term in the following sequence?
::以下顺序的第10个学期是什么?

$\begin{aligned} - 12, 6, - 3, \frac{3}{2}, \dots \end{aligned}$

Solution:
::解决方案 :

The sequence is geometric and the common ratio is $\begin{aligned} - \frac{1}{2} \end{aligned}$ . The equation is $\begin{aligned} a_{n} = - 12 \cdot {(- \frac{1}{2})}^{n - 1} \end{aligned}$ . The 10th term is
::顺序是几何,共同比率是-12。方程式是 12(--12)n-1。第十个词是

$\begin{aligned} - 12 \cdot {(- \frac{1}{2})}^{9} = \frac{3}{128} . \end{aligned}$

Example 6
::例6

What is the 10th term in the following sequence?
::以下顺序的第10个学期是什么?

$\begin{aligned} - 1, \frac{2}{3}, \frac{7}{3}, 4, \frac{17}{3}, \dots \end{aligned}$

Solution:
::解决方案 :

The pattern might not be immediately recognizable, but try ignoring the $\begin{aligned} \frac{1}{3} \end{aligned}$ in each number to see the pattern a different way.
::模式也许无法立即识别,但试图忽略每个数字中的13个,以不同的方式看待模式。

$\begin{aligned} - 3, 2, 7, 12, 17, \dots \end{aligned}$

You should see the common difference of 5. This means the common difference from the original sequence is $\begin{aligned} \frac{5}{3} \end{aligned}$ . The equation is $\begin{aligned} a_{n} = - 1 + (n - 1) (\frac{5}{3}) \end{aligned}$ . The 10th term is
::您应该看到5之间的共同区别。这意味着与原始序列的相同区别是53, 方程式是 an1+1 (53)。第十个词是

$\begin{aligned} - 1 + 9 \cdot (\frac{5}{3}) = - 1 + 3 \cdot 5 = - 1 + 15 = 14. \end{aligned}$

Example 7
::例7

Find an equation that defines the $\begin{aligned} a_{k} \end{aligned}$ term for the following sequence:
::为以下序列查找一个方程式来定义ak 术语:

$\begin{aligned} 0, 3, 8, 15, 24, 35, \dots \end{aligned}$

Solution:
::解决方案 :

The sequence is neither arithmetic nor geometric. It will help to find the pattern by examining the common differences, and then the common differences of the common differences. This numerical process is connected to ideas in calculus.
::序列既不是算术,也不是几何。它有助于通过研究共同差异找到模式,然后研究共同差异的共同差异。这个数字过程与微积分中的想法相关联。

0, 3, 8, 15, 24, 35

3, 5, 7, 9, 11

2, 2, 2, 2

Notice that when you examine the common differences of the common differences, the pattern becomes increasingly clear. Since it took two layers to find a constant function, this pattern is quadratic and very similar to the perfect squares.
::请注意, 当检查共同差异的共同差异时, 模式会变得越来越清晰。由于需要两层才能找到一个恒定的函数, 这种模式是四方形的, 与完美的正方形非常相似。

$\begin{aligned} 1, 4, 9, 16, 25, 36, \dots \end{aligned}$

The $\begin{aligned} a_{k} \end{aligned}$ term can be described as $\begin{aligned} a_{k} = k^{2} - 1 \end{aligned}$ .
::ak 术语可描述为ak=k2-1。

Summary
::摘要

A sequence is a list of numbers separated by commas.
::序列是用逗号分隔的数字列表。

The common pattern in an arithmetic sequence is that the same number is added or subtracted to each number to produce the next number. This is called the common difference.
::算术序列中的常见模式是,为产生下一个数字,在每个数字中加上或减去相同数字,这称为共同差数。

The common pattern in a geometric sequence is that the same number is multiplied or divided to each number to produce the next number. This is called the common ratio.

::几何序列中的常见模式是,同一数字乘以或除以每个数字,得出下一个数字。这称为共同比率。

Review
::回顾

Use the sequence $\begin{aligned} 1, 5, 9, 13, \dots \end{aligned}$ for questions 1-3.
::使用顺序 1,5,9,13,... 用于问题1,3

1. Find the next three terms in the sequence.
::1. 在顺序中找到下三个术语。

2. Find an equation that defines the $\begin{aligned} a_{k} \end{aligned}$ term of the sequence.
::2. 找到一个方程式,界定序列的方程的方程。

3. Find the 150th term of the sequence.
::3. 找出序列的第150个术语。

Use the sequence $\begin{aligned} 12, 4, \frac{4}{3}, \frac{4}{9}, \dots \end{aligned}$ for questions 4-6.
::使用顺序12,4,43,49... 用来回答问题4 -6

4. Find the next three terms in the sequence.
::4. 在顺序中找到下三个术语。

5. Find an equation that defines the $\begin{aligned} a_{k} \end{aligned}$ term of the sequence.
::5. 找到一个方程式,确定序列的方程的方程。

6. Find the 17th term of the sequence.
::6. 找出序列的第17个术语。

Use the sequence $\begin{aligned} 10, - 2, \frac{2}{5}, - \frac{2}{25}, \dots \end{aligned}$ for questions 7-9.
::使用顺序10,-2,25,-225,-225,7 -9。

7. Find the next three terms in the sequence.
::7. 在顺序中找到下三个术语。

8. Find an equation that defines the $\begin{aligned} a_{k} \end{aligned}$ term of the sequence.
::8. 找出一个方程,界定序列的方程的方程。

9. Find the 12th term of the sequence.
::9. 找出顺序的第12个术语。

Use the sequence $\begin{aligned} \frac{7}{2}, \frac{9}{2}, \frac{11}{2}, \frac{13}{2}, \dots \end{aligned}$ for questions 10-12.
::使用顺序72,92,112,132 来回答问题10 -12

10. Find the next three terms in the sequence.
::10. 在顺序中找到下三个术语。

11. Find an equation that defines the $\begin{aligned} a_{k} \end{aligned}$ term of the sequence.
::11. 找出一个方程式,界定序列的方程式用方程式定义的方程式。

12. Find the 314th term of the sequence.
::12. 找出序列的第314个术语。

13. Find an equation that defines the $\begin{aligned} a_{k} \end{aligned}$ term of the sequence $\begin{aligned} 4, 11, 30, 67, \dots \end{aligned}$
::13. 找出一个方程式来定义序列 4,11,30,67的方程的方程

14. Explain the connections between arithmetic sequences and linear functions.
::14. 解释算术序列和线性函数之间的联系。

15. Explain the connections between geometric sequences and exponential functions.
::15. 解释几何序列和指数函数之间的联系。

16. An ant colony invades the caramels in a candy store. The 1st day they eat $\begin{aligned} \frac{1}{4} \end{aligned}$ of a caramel, the 2nd day $\begin{aligned} \frac{1}{2} \end{aligned}$ of a caramel, and the 3rd day $\begin{aligned} \frac{3}{4} \end{aligned}$ .
::16. 蚂蚁聚居地入侵糖果店的焦糖,第一天吃14焦糖,第二天吃12焦糖,第三天吃34。

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

b) How many do you think they’ll eat on the 4th, 5th, and 6th days?
:b) 你认为他们将在第四、第五和第六天吃多少?

17. On the way home from school on the day of a trip downtown, a bunch of students stopped off at an arcade. Sam and Henry began to play one of their favorite games, which features aliens. "In this video game," Sasha stated convincingly, "an alien splits into two aliens, who then split into two more aliens every 10 minutes." Henry challenged Sasha, "So tell me how many aliens there would be after they split 10 times." What is the answer to Henry's challenge?
::17. 在市区旅行当天从学校回家的路上,一群学生在游乐场停了下来,Sam和Henry开始玩他们最喜欢的游戏之一,其中有一个是外国人。萨沙令人信服地说,“在这个游戏中,一个外国人分裂成两个外国人,然后每10分钟又分裂成两个外国人。”Henry挑战Sasha,“告诉我在他们分裂10次之后有多少外国人会出现。” 亨利挑战的答案是什么?

18. The amount of memory that computer chips can hold in the same amount of space doubles every year. In 1992 they could hold 1MB. Determine the amount of memory on the same size chip in 2007.
::18. 计算机芯片每年可以保持的存储量与空间的两倍相等,1992年可以保持1MB,2007年确定同一大小芯片的存储量。

Review (Answers )
::回顾(答复)

Please see the Appendix.
::请参看附录。

章节大纲

Introduction ::导言

Arithmetic Sequence ::亚学序列

Geometric Sequence ::几何序列

Examples ::实例

Summary ::摘要

Review ::回顾

Review (Answers ) ::回顾(答复)

Introduction
::导言

Arithmetic Sequence
::亚学序列

Geometric Sequence
::几何序列

Examples
::实例

Summary
::摘要

Review
::回顾

Review (Answers )
::回顾(答复)