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雙語教材 1 數列與級數 Sequences and Series
數列 Sequences
1. 認識數列 Understand sequences
2. 等差數列 Arithmetic sequences
3. 等比數列 Geometric sequences
等差級數 Arithmetic series
1. 等差級數的和 Sum of a finite arithmetic series
2. 等差級數和的公式 Formula for the sum of a finite arithmetic series
核心單字
英文
參考音標
數列
Sequence
/ˈsiː.kwəns/
級數
Series
/ˈsɪr.iːz/
數列的項
Term of a squence
等差數列
Arithmetic sequence
/əˈrɪθ.mə.tɪk/
等差級數
Arithmetic series
公差
Common difference
/ˈkɑː.mən/ /ˈdɪf.ɚ.əns/
等比數列
Geometric sequence
/ˌdʒiː.əˈmet.rɪk/
等比級數
Geometric series
公比
Common ratio
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數列 Sequence
1 認識數列
Understand sequences
數列英文版定義：
Definition: A
number sequence
is a pattern of numbers arranged in a particular
order according to a rule.
Each individual number, or element, of a sequence is called a
term.
課本內容：
雙語使用參考範例：
The right table(右表) is the schedule of Bus F216. The departure time of the
North third gate of Taipei station is 11, 12, 13, 14, 15, 16, 21 and 22 o'clock.
Numbers arranged in a row like this and separated by commas(逗號) are
called sequences.
Here are some number sequences.
2, 4 , 6, 8, …
100, 400, 1600, 6400,…
1 , 4, 9, 16, 25,…
3, 7, 10, 12, 13, …
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課本內容：
在上面的數列中：
11 稱為這個數列的第 1 項或
首項 (First term)
，通常記為 a
1
；
12 稱為這個數列的第 2 項(Second term)，記為 a
2
；
13 稱為這個數列的第 3 項(Third term)，記為 a
3
；
這個數列的第 n 項，記為 a
n
；而數列中的最後一項稱為
末項 (Last term)
。
翻譯示例：
In the above sequence: 11 is called the first term of this sequence, which is usually
recorded as a
1
.
雙語使用參考範例：
You can also use the notation a
n
to denote the nth term of a sequence, where n
is a positive integer.
如何念
a
n
T: How to say
a
n
: a subscript n or a sub n
Remember that n, the term number, will always be a whole number.
We could not have a 0.25th term, or a
1
4
th term.
備註：
基數
序數
唸法
寫法
1～3
不規則變化
first、second、third
1st、2nd、3rd
4～19
字尾加 th
* ve 結尾改成 th
* t 結尾直接加 h
* e 結尾去 e 加 th
fourth,fifth,sixth,
seventh,eighth,nont
h...
數字後加上 th。
（例：4th, 20th, 100th）
20, 30,
40...
字尾去 y 加 ieth
twentieth
thirtieth
fortieth
100,
1000...
字尾加 th
hundreth,
thousandth,
millionth
21～99
（0 結尾以
外）
個位數改為序數，
中間加連字號（-）
twenty-first,
thirty-fourth,
seventy-second
數字後加上 th，
但個位數為 1～3 者除
外，
1～3 分別加上 st, nd,
rd。
（例：21st、32nd、43rd）
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雙語使用參考範例：
在數列2, 4, 6, 8, 10, 12, 14中
哪一個是首項? T: Which one is the first term?
6是第幾項? T: Which term is 6? & 6 is the ______ term? S: Third
這個數列總共有幾項? T: How many terms are there in this sequence?
請找出/寫下前三項 T: Please find/write down the first three terms in this
sequence.
練習
數列 1, 4, 7, 10, 13, 16 中，
第 1 項 a
1
＝
1
，第 2 項 a
2
＝
4
，第 3 項 a
3
＝
7
，
第 4 項 a
4
＝
10
，第 5 項 a
5
＝
13
，第 6 項 a
6
＝
16
。
翻譯示例：
In the sequence 1, 4, 7, 10, 13, and 16,
The first term a
1
= 1, the second term a
2
= 4, the third term a
3
= 7,
the fourth term a
4
= 10, the fifth term a
5
= 13,and the sixth term a
6
= 16.
課本內容:
數列可能具有某種規律。例如：
翻譯示例：
The sequence may have some mathematical patterns. For example,
可以這麼說：
我們可以使用規律來…規律可以幫助我們…
We can use patterns to… / Patterns can help us…
make predictions
（預測）about the future and calculate how long it will take for
an investment to double in value.
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觀察規律：
數列如下：
135, 137, 139, 141, ……，
可以發現這個數列的規律是：相鄰兩項中，後一項等於前一項加 2。
135 , 137 , 139 , 141 , ……
＋2 ＋2 ＋2 ＋2
翻譯示例：
The address numbers of the shops on the street are as follows:
135, 137, 139, 141, ……，
It can be found that the rule of this sequence is: among the two adjacent terms, the
latter term is equal to the previous term plus 2.
*鄰近的 adjacent /əˈdʒeɪ.sənt/
練習
已知下列各數列分別隱含某種規律，依其規律在空格中填入適當的數。
It is known that each of the following series implies a certain pattern.
Please fill in the appropriate number in the space according to its pattern.
(1) 10, 18, 26,
, 42, 50
(2) 2, 4, 8,
, 32, 64
翻譯示例：
It is known that each of the following series implies a certain pattern.
Please fill in the appropriate number in the space according to its pattern.
*適當的 appropriate /əˈproʊ.pri.ət/
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課本內容:
有些具有規律的數列，其第 n 項 a
n
可用含有未知數 n 的式子表示，例如：
翻譯示例：
For some regular sequences, the nth term a
n
can be expressed by the formula
containing the unknown n. For example:
(1) 從 1 開始的連續正整數的
倒數所形成的數列如下：
1
1
,
1
2
,
1
3
,
1
4
,……
第
1
項 第
2
項 第
3
項
第
n
項
a
1
a
2
a
3
……
a
n
1
1
1
2
1
3
1
n
翻譯示例：
The sequence formed by the reciprocal of consecutive positive integers starting
from 1 is as follows:
1
1
,
1
2
,
1
3
,
1
4
,……
*倒數 reciprocal /rɪˈsɪp.rə.kəl/
*連續的 consecutive /kənˈsek.jə.t̬ɪv/
*正整數 positive integers
(2) 從 2 開始的連續正偶數所
形成的數列如下：
2, 4, 6, 8,……
第
1
項 第
2
項 第
3
項
第
n
項
a
1
a
2
a
3
……
a
n
2
4
6
2n
2
×
1
2
×
2
2
×
3
2
×
n
翻譯示例：
The sequence formed by consecutive positive even numbers starting from 2 is as
follows: 2, 4, 6, 8,……
*偶數 even numbers
*奇數 odd numbers
像這類具有規律的數列，如果能寫出第 n 項的表示式，就可以找到該數列
的任意一項。上述例
(2)
中，第 n 項 a
n
＝2n 稱為此數列的
一般項
。
英文版定義：
Definition: If we can write the expression of the nth term in a regular sequence like
this, we can find any term of the sequence. In the above example (2), the nth term
a
n
＝2n is called the
general term
of the sequence.
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可以這麼說：
如何找出一般項 T: How to find the general term?
一般項為何? T: What is the general term?
請寫下這個數列的一般項 T: Write a general term of the sequence.
你觀察/注意到甚麼? T: What did you observe/notice?
觀察一般項：2, 4, 6, 8,……
T: You can notice that the value of each term is two times n.
觀察一般項：8, 11, 14, 17, 20, 23, 26,……
T: You can notice that the value of each term is three bigger than the value of
the previous term.
若要求第 100 項 a
100
，只要在 a
n
＝2n 的式子中，n 以 100 代入即可求得：
A
100
＝2
×
100＝200。
翻譯示例：
If we are asked to find a
100
, as long as n is substituted with 100 in the formula a
n
＝
2n.
A
100
＝2
×
100＝200
*用…代替 substitute /ˈsʌb.stə.tuːt/
例題 求第 n 項
已知某數列的第 n 項 a
n
＝n﹙n＋1﹚﹙n＋2﹚，
(1) 求此數列的前 3 項。
(2) 求此數列的第 10 項。
(3) 若此數列的第 k 項為 336，求 k。
翻譯示例：
Find nth term
Given the nth term of a sequence, a
n
＝n﹙n＋1﹚﹙n＋2﹚.
(1) Find the first three terms of this sequence.
(2) Find the 10
th
term of this sequence.
(3) If the kth term of this sequence is 336, find k.
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參考資料來源
1. 110 國中數學 2 下翰林版課本
2. IB Maths SL Book Oxford
Chapter 6 Patterns, sequences andseries
3. Holt McDougal Larson Algebra 2
Chapter 7 Sequences and Series
☆老師們可以自己從中選擇以做出適合自己學生程度的學習單或是在課堂中適
時補充這些英文。
製作者：臺北市立誠正國民中學 陳怡伶