Course: 7.2 明确公式 | The Way To Learn

Section outline

Select section General

Collapse Expand
General

Collapse all Expand all
- Select activity 新闻通告
  
  新闻通告 Forum

明确公式

Rachel and Elaina have started a website where they debate the best color of hair dye. The show is really popular and the visitors to their website are increasing very rapidly. They figure that membership is increasing by about 500 people every three days.
::瑞秋和伊莱娜启动了一个网站,讨论染发最美的颜色。节目非常受欢迎,访问网站的人也迅速增加。他们认为会员人数每三天增加500人。

At this rate, how many members will they have on the 48th day? How many days will it be before they reach 25,000 members?
::照此速度,第48天有多少成员? 还要多少天才能达到25 000名成员?

Explicit Formulas
::明确公式

When we represent a with a formula that lets us find any term in the sequence without knowing any other terms, we are representing the sequence explicitly.
::当我们代表一个公式,让我们在序列中找到任何术语而不知道任何其他术语时,我们明确地代表了顺序。

Given a recursive definition of an arithmetic or geometric sequence, you can always find an explicit formula, or an equation to represent the n ^th term of the sequence. Consider for example the sequence of odd numbers we started with: 1,3,5,7,...
::根据对算术或几何序列的递归定义, 您总是可以找到一个明确的公式, 或者代表序列 nth 术语的方程式。例如, 考虑我们以 1, 3, 5, 7,... 开始的奇数序列 : 1, 3, 5, 7,...

We can find an explicit formula for the n ^th term of the sequence if we analyze a few terms:
::如果我们分析几个术语,我们就可以为序列的第n期找到一个明确的公式:

a ₁ = 1
::a 1 = 1

a ₂ = a ₁ + 2 = 1 + 2 = 3
::a2 = a1 + 2 = 1 + 2 = 2 = 3

a ₃ = a ₂ + 2 = 1 + 2 + 2 = 5
::a3=a2+2=1+2+2+2=5

a ₄ = a ₃ + 2 = 1 + 2 + 2 + 2 = 7
::a4=a3+2=1+2+2+2+2=7

a ₅ = a ₄ + 2 = 1 + 2 + 2 + 2 + 2 = 9
::a5=a4+2=1+2+2+2+2+2+2+2=9

a ₆ = a ₅ + 2 = 1 + 2 + 2 + 2 + 2 + 2 = 11
::a6=a5+2=1+2+2+2+2+2+2+2+2+2+2+2=11

Note that every term is made up of a 1, and a set of 2’s. How many 2’s are in each term?
::请注意,每个学期由一和二组成。每个学期有多少学期?

a ₁	= 1
a ₂	= 1 + 2 = 3
a ₃	= 1 + 2 × 2 = 5
a ₄	= 1 + 3 × 2 = 7
a ₅	= 1 + 4 × 2 = 9
a ₆	= 1 + 5 × 2 = 11

The n ^th term has ( n - 1) 2's. For example, a ₉₉ = 1 + 98 × 2 = 197 . We can therefore represent the sequence as a _n = 1 + 2( n - 1). We can simplify this expression:
::Nth 术语为 (n - 1) 2 。例如, a99 = 1 + 98 × 2 = 197。因此, 我们可以将序列表示为 = 1 + 2 (n - 1) 。我们可以简化此表达式 :

a _n	= 1 + 2( n - 1)
a _n	= 1 + 2 n - 2
a _n	= 2 n - 1

In general, we can represent an arithmetic sequence in this way, as long as we know the first term and the common difference, d . Notice that in the previous example, the first term was 1, and the common difference, d , was 2. The n ^th term is therefore the first term, plus d ( n - 1):
::一般而言,只要我们知道第一个任期和共同差异,我们就可以以这种方式代表一个算术序列。 d. 注意到在前一例中,第一个任期为1, 共同差异为2。因此,第一个任期为N,加上d-1:

a _n	= a ₁ + d ( n - 1)

You can use this general equation to find an explicit formula for any term in an arithmetic sequence.
::您可以使用此通用方程式在算术序列中为任何术语找到一个明确的公式。

Examples
::实例

Example 1
::例1

Earlier, you were asked two questions about a website membership.
::早些时候,有人问你们两个关于网站成员的问题。

If memberships are increasing by about 500 people every three days, how many members will they have on the 48th day? How many days will it be before they reach 25,000 members?
::如果成员每三天增加约500人,那么在第48天他们有多少成员? 在他们达到25 000名成员之前还要多少天?

This is actually a fairly simple arithmetic sequence: each day there are 500/3 more members, on average. Use the formula for from the Example 2 below.
::实际上,这是一个相当简单的算术序列:每天平均有500/3的成员。使用下面例2中的公式。

Example 2
::例2

Find an explicit formula for the nth term of the sequence 3, 7, 11, 15... and use the equation to find the 50 ^th term in the sequence.
::为序列3、7、11、15的第n期寻找一个明确的公式并利用方程式在序列中找到第50期

a _n = 4 n - 1 , and a ₅₀ = 199
::a=4n-1和a50=199

The first term of the sequence is 3, and the common difference is 4.
::顺序的第一个任期是3年,共同的区别是4年。

a _n	= a ₁ + d ( n - 1)
a _n	= 3 + 4( n - 1)
a _n	= 3 + 4 n - 4
a _n	= 4 n - 1

a ₅₀	= 4(50) - 1 = 200 - 1 = 199

We can also find an explicit formula for a geometric sequence. Consider the following sequence:
::我们还可以为几何序列找到一个明确的公式。考虑以下顺序:

		t ₂ = 2 t ₁ = 2 × 3 = 6
t ₁ = 3	$\begin{aligned} \to \end{aligned}$	t ₃ = 2 t ₂ = 2 × 6 = 12
t _n = 2 × t _n _-1		t ₄ = 2 t ₃ = 2 × 12 = 24
		t ₅ = 2 t ₄ = 2 × 24 = 48

Notice that every term is the first term, multiplied by a power of 2. This is because 2 is the common ratio for the sequence.
::注意每个学期都是第一个学期,乘以2的功率,这是因为2是序列的共同比率。

t ₁	= 3
t ₂	= 2 × 3 = 6
t ₃	= 2 × 2 × 6 = 2 ² × 6 = 24
t ₄	= 2 × 2 × 2 × 6 = 2 ³ × 6 = 48
t ₅	= 2 × 2 × 2 × 2 × 6 = 2 ⁴ × 6 = 96

The power of 2 in the n ^th term is ( n -1). Therefore the n ^th term in this sequence can be defined as: t _n = 3(2 ⁿ ^{- 1} ). In general, we can define the n ^th term of a geometric sequence in terms of its first term and its common ratio, r :
::第n- 1 项中的 2 功率是 (n-1) 。因此,此序列中的 n 值可以定义为: tn = 3( 2n-1 ) 。一般来说, 我们可以从第一个术语及其共同比率( r) 中来定义几何序列的 n 值 :

t _n	= t ₁ (r ⁿ ^-1 )

You can use this general equation to find an explicit formula for any term in a geometric sequence.
::您可以使用这个通用方程来为几何序列中的任何术语找到一个明确的公式。

Example 3
::例3

Find an explicit formula for the n ^th term of the sequence 5, 15, 45, 135... and use the equation to find the 10 ^th term in the sequence.
::为顺序5、15、45、135的 nth 术语寻找明确的公式并利用方程式在序列中找到第10个术语

a _n = 5 × 3 ⁿ ^{- 1} , and a ₁₀ = 98,415
::a = 5 × 3n-1,和 a10 = 98 415

The first term in the sequence is 5, and r = 3.
::顺序的第一个学期是5个学期,r=3个学期。

a _n	= a ¹ × r ⁿ ^{- 1}
a _n	= 5 × 3 ^{n - 1}
a ₁₀	= 5 × 3 ^{10 - 1}
a ₁₀	= 5 × 3 ⁹ = 5 × 19,683 = 98,415

Again, it is always possible to write an explicit formula for terms of an arithmetic or geometric sequence. However, you can also write an explicit formula for other sequences, as long as you can identify a pattern. To do this, you must remember that a sequence is a function, which means there is a relationship between the input and the output. That is, you must identify a pattern between the term and its index , or the term’s “place” in the sequence.
::再次,总是有可能为计算或几何序列的术语写出明确的公式。然而,只要您能够识别一个模式,您也可以为其他序列写出明确的公式。要做到这一点,您必须记住,一个序列是一个函数,这意味着输入和输出之间存在某种关系。也就是说,您必须确定术语与其索引或序列中的术语“位置”之间的一种模式。

Example 4
::例4

Write an explicit formula for the nth term of the sequence 1, (1/2), (1/3), (1/4)...
::为序列 1 (1/2)、 (1/3)、 (1/4) n 的 n 术语写一个明确的公式...

a _n = (1/ n )
::a = (1/n) = (1/n)

Initially you may see a pattern in the fractions, but you may also wonder about the first term. If you write 1 as (1/1), then it should become clear that the n ^th term is (1/ n ).
::起初,您可能会看到分数中的模式, 但是您也可能对第一个学期感到疑惑。如果您将 1 写为 1 (1/1), 那么应该可以清楚地看到 n 学期是 1/ n 。

Example 5
::例5

Write an explicit formula for the sequence: 2, 9, 16... and use the formula to find the value of the 20 ^th term.
::为顺序写一个明确的公式: 2, 9, 16... 并使用公式来找到第20个学期的价值。

For the sequence: 2, 9, 16...
::顺序: 2 9 16...

$\begin{aligned} a_{n} = 7 n - 5 \end{aligned}$
::an=7n- 5

$\begin{aligned} ∴ a_{20} = 7 (20) - 5 \end{aligned}$
::=a20=7(20)-5

$\begin{aligned} a_{20} = 135 \end{aligned}$
::a20=135

Example 6
::例6

Write an explicit formula for the sequence: (1/2), (1/4), (1/8) and use the formula to find the value of the 7 _th term.
::为顺序写一个明确的公式1/2)、(1/4)、(1/8),并使用公式查找第7个任期的价值。

For the sequence: (1/2), (1/4), (1/8)...
::顺序:1/2)、1/4、1/8.

$\begin{aligned} a_{n} = \frac{1}{2^{n}} \end{aligned}$
::an=12n an=12n

$\begin{aligned} ∴ a_{7} = \frac{1}{2^{7}} \end{aligned}$
::*a7=127

$\begin{aligned} a_{7} = \frac{1}{128} \end{aligned}$
::a7=1128 (a7=1128)

Example 7
::例7

Identify all sequences in the previous two examples that are geometric. What is the common ratio in each sequence?
::前两个示例中的所有序列均为几何。每个序列的共同比率是多少?

The sequence in Example 5 is arithmetic.
::例5中的顺序是算术。

The sequence in Example 6 is geometric and has r = 1/2.
::例6中的顺序是几何,R=1/2。

Review
::回顾

Name the sequence as arithmetic, geometric, or neither.
::将序列命名为算术、几何或两者无一。

$\begin{aligned} - 21, - 6, 18, - 3, 20, - 2 \end{aligned}$
$\begin{aligned} 0, \frac{- 1}{5}, \frac{- 2}{5}, \frac{- 3}{5}, \frac{- 4}{5}, - 1 \end{aligned}$
$\begin{aligned} 1, 3, 9, 27, 81, 243 \end{aligned}$
$\begin{aligned} 2, 9, - 2, 1, 18, 2 \end{aligned}$

Write the first 5 terms of the arithmetic sequence (explicit).
::写入算术序列( 明确) 的前 5 个条件。

$\begin{aligned} a_{n} = - 8 - 9 (n - 1) \end{aligned}$
::an8-9(n-1)

$\begin{aligned} a_{n} = 6 - \frac{2}{3} (n - 1) \end{aligned}$
::a=6-23(n-1)

$\begin{aligned} a_{n} = 8 + \frac{1}{3} (n - 1) \end{aligned}$
::a=8+13(n- 1)

Solve the following:
::解决以下问题:

What are the first five terms of the sequence? $\begin{aligned} a_{n} = a_{n - 1} - \frac{10}{3}; a_{1} = - 6 \end{aligned}$
::序列的前五个条件是什么? an=-1--103;a1 @%6

Given the sequence, write a recursive function to generate it: $\begin{aligned} 2, - 4, - 10, - 16, - 22, - 28 \end{aligned}$
::根据顺序,写一个递归函数来生成它: 2 -4, - 10, - 16 - 22, - 28

Write the equation of $\begin{aligned} a_{n} \end{aligned}$ without using recursion: $\begin{aligned} a_{n} = a_{n - 1} - \frac{3}{2}; a_{1} = 10 \end{aligned}$
::不使用递转 : an= an - 1 - 32; a1= 10

Write as a recursion: $\begin{aligned} a_{n} = 6 - \frac{5}{3} (n - 1) \end{aligned}$
::以递归书写 : an=6- 53(n- 1)

Write the equation of $\begin{aligned} a_{n} \end{aligned}$ without using recursion: $\begin{aligned} a_{n} = a_{n - 1} + 8; a_{1} = 3 \end{aligned}$
::在不使用递转 : an= an- 1+8; a1=3 的情况下写入 a 的方程式

What are the first five terms of the sequence? $\begin{aligned} a_{n} = a_{n - 1} - 1; a_{1} = - 5 \end{aligned}$
::序列的前五个条件是什么? a=-1-1;a1+5

Write an explicit formula for the $\begin{aligned} n_{t h} \end{aligned}$ term of the arithmetic sequence.
::为算术序列 nth 术语写一个明确的公式。

$\begin{aligned} - 7, \frac{- 13}{3}, \frac{- 5}{3}, 1, \frac{11}{3}, \frac{19}{3} \end{aligned}$
$\begin{aligned} 6, - 4, - 14, - 24, - 34, - 44 \end{aligned}$
$\begin{aligned} 9, 16, 23, 30, 37, 44 \end{aligned}$

In a particular arithmetic sequence, the second term is 4 and the fifth term is 13. Write an explicit formula for this sequence.
::在一个特定的算术序列中,第二个任期为4个,第五个任期为13个。为这个序列写一个明确的公式。

Write the first 5 terms of the geometric sequence.
::写下几何序列的前五个条件

$\begin{aligned} a_{n} = 5 (- 3)^{(n - 1)} \end{aligned}$
::an=5(- 3)(n-1)

$\begin{aligned} a_{n} = - 6 (- \frac{10}{3})^{n - 1} \end{aligned}$
::an*6(- 103)n- 1

Write the explicit formula for the $\begin{aligned} n_{t h} \end{aligned}$ term of the geometric sequence.
::写入几何序列 nth 术语的明确公式。

$\begin{aligned} - 8, 16, - 32, 64, - 128, 256 \end{aligned}$
$\begin{aligned} 9, \frac{27}{2}, \frac{81}{4}, \frac{243}{8}, \frac{729}{16}, \frac{2187}{32} \end{aligned}$

Convert the explicit formula to a recursive formula.
::将明确的公式转换成递归公式。

$\begin{aligned} a_{n} = 9 (\frac{- 4}{3})^{(n - 1)} \end{aligned}$
::an=9(- 43)(n-1)

$\begin{aligned} a_{n} = - 6 (- 4)^{(n - 1)} \end{aligned}$
::an6(- 4)(n-1)

$\begin{aligned} a_{n} = - 5 (5)^{(n - 1)} \end{aligned}$
::an5(5)(n-1)

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

General

明确公式

Explicit Formulas ::明确公式

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Example 6 ::例6

Example 7 ::例7

Review ::回顾

Review (Answers) ::回顾(答复)