Section: 亚学和几何序列 | 12.2 测深和几何序列

Section outline

A is a list of numbers with a common pattern. The common pattern in an arithmetic sequence is that the same number is added or subtracted to each number to produce the next number. The common pattern in a geometric sequence is that the same number is multiplied or divided to each number to produce the next number.
::A 是一个具有共同模式的数字列表。计算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,以产生下一个数字。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。

Are all sequences arithmetic or geometric?
::所有序列都是算术还是几何?

Sequences
::序列

A sequence is just a list of numbers separated by commas. A sequence can be finite or infinite. If the sequence is infinite, the first few terms are followed by an ellipsis $\begin{aligned} (\dots) \end{aligned}$ indicating that the pattern continues forever.
::序列只是用逗号分隔的数字列表。序列可以是有限的, 也可以是无限的。如果序列是无限的, 最初几个词之后是省略号 , 表示该图案会永远持续下去。

An infinite sequence : $\begin{aligned} 1, 2, 3, 4, 5, \dots \end{aligned}$
::无限序列: 1,2,3,4,5,...

A finite sequence: $\begin{aligned} 2, 4, 6, 8 \end{aligned}$
::限定序列:2,4,6,8

In general, you describe a sequence with subscripts that are used to index the terms. The $\begin{aligned} k^{t h} \end{aligned}$ term in the sequence is $\begin{aligned} a_{k} \end{aligned}$ .
::一般而言,您描述一个带有下标的序列,用于对术语进行索引。该序列中的 kth 术语为ak。

$\begin{aligned} a_{1}, a_{2}, a_{3}, a_{4}, \dots, a_{k}, \dots \end{aligned}$
::a1,a2,a3,a4,... ... ak,...

Arithmetic Sequences
::亚学序列

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和共同差数定义。

\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

Geometric Sequences
::几何序列

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和一个共同比率器定义。

\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

When trying to determine what kind of sequence it is, first test for a common difference and then test for a common ratio. If the sequence does not have a common difference or ratio, it is neither an arithmetic or geometric sequence.
::当试图确定它属于哪种序列时,首先测试共同差异,然后测试共同比率。如果该序列没有共同差异或比率,那么它既不是一个算术或几何序列。

$\begin{aligned} 0.135, 0.189, 0.243, 0.297, \dots \end{aligned}$ is an arithmetic sequence because the common difference is 0.054.
::0.135,0.189,0.243,0.297, 是一个算术序列, 因为共同差数是0.054。

$\begin{aligned} \frac{2}{9}, \frac{1}{6}, \frac{1}{8}, \dots \end{aligned}$ is a geometric sequence because the common ratio is $\begin{aligned} \frac{3}{4} \end{aligned}$ .
::29,16,18,... 是一个几何序列因为共同比率是34

$\begin{aligned} 0.54, 1.08, 3.24, \dots \end{aligned}$ 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.
::0.54,1.08,3.24.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。

Examples
::实例

Example 1
::例1

Earlier, you were asked if all sequences are arithmetic or geometric. The sequence above shows that not all sequences are arithmetic or geometric. Two famous sequences that are neither arithmetic nor geometric are the Fibonacci sequence and the sequence of prime numbers.
::早些时候,有人问您是否所有序列都是算术或几何。上面的序列显示并非所有序列都是算术或几何。两个既非算术也非几何的著名序列是Fibonacci序列和质数序列。

Fibonacci Sequence: $\begin{aligned} 1, 1, 2, 3, 5, 8, 13, 21, 34, \dots \end{aligned}$
::Fibonacci序列: 1,1、2,3,5,8,13,21,34,...

Prime Numbers: $\begin{aligned} 2, 3, 5, 7, 11, 13, 17, 19, 23, \dots \end{aligned}$
::总理号: 2,3,5,7,11,13,17,19,23...

Example 2
::例2

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,以及下三个任期。

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

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. Always check the sequence in multiple places to make sure that the common difference holds up throughout.
::这个序列既不是算术,也不是几何。最初几个术语之间的差异是- 29, 29, 1081, 14243。虽然最初存在共同差异, 但这一差异并没有贯穿于序列中。总是要检查多个地方的序列, 以确保共同差异始终保持。

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 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,但仍有可能通过注意到分子的计算顺序是计算顺序,其起始任期为2个,共同差别为2个。分母是几何序列,初始任期为3个,共同比例为3个。

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

Example 3
::例3

What is the tenth term in the following sequence?
::以下顺序中的第十个学期是什么?

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

The sequence is geometric and the common ratio is $\begin{aligned} - \frac{1}{2} \end{aligned}$ . The equation is:
::顺序是几何,共同比率是-12。方程式是:

$\begin{aligned} a_{n} = - 12 \cdot {(- \frac{1}{2})}^{n - 1} \end{aligned}$ .
::an*12(- 12)- (- 12)n- ( 1) 。

The tenth term is:
::第十个任期是:

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

Example 4
::例4

What is the tenth term in the following sequence?
::以下顺序中的第十个学期是什么?

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

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 $\begin{aligned} 10^{t h} \end{aligned}$ term is:
::您应该看到5之间的共同差别。这意味着与原始序列的相同区别是53, 方程式是 an1+(n- 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 5
::例5

Find an equation that defines the $\begin{aligned} a_{k} \end{aligned}$ term for the following sequence.
::查找一个方程式, 用于定义以下序列的 ak 术语。

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

The sequence is not 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.
::序列不是算术,也不是几何。它有助于通过检查共同差异和共同差异的共同差异来找到模式。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 when you examine the common difference 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.
::当您检查共同差异的共同差异时注意, 模式变得越来越清晰。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

A sequence is a list of numbers with a common pattern, which can be finite or infinite.
::一个序列是带有共同模式的数字列表,可以是有限或无限的。

Arithmetic sequences are defined by an initial value and a common difference, with the same number added or subtracted to each term.
::相对序列的定义是初始值和常见差异,每个术语增加或减去相同数字。

Geometric sequences are defined by an initial value and a common ratio, with the same number multiplied or divided to each term.
::几何序列由初始值和共同比率来界定,同一数字乘以或除以每个术语。

To determine the type of sequence, first test for a common difference, then test for a common ratio.
::为确定序列类型,首先测试共同差异,然后测试共同比率。

If a sequence does not have a common difference or ratio, it is neither an arithmetic nor a geometric sequence.
::如果一个序列没有共同的差异或比率,它既不是算术,也不是几何序列。

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 $\begin{aligned} 150^{t h} \end{aligned}$ 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 $\begin{aligned} 17^{t h} \end{aligned}$ 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 $\begin{aligned} 12^{t h} \end{aligned}$ 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 $\begin{aligned} 314^{t h} \end{aligned}$ 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 for the sequence $\begin{aligned} 4, 11, 30, 67, \dots \end{aligned}$
::13. 找出一个方程式,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。

Review (Answers)
::回顾(答复)

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。

Section outline

Sequences ::序列

Arithmetic Sequences ::亚学序列

Geometric Sequences ::几何序列

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)