数列,等差、等比数列求和公式 | 玄数

2012-01-24

1.   数列

按照一定顺序排列着的一列数称为数列Sequence of Number),数列中的每一个数叫做这个数列的(Members / Elements / Iterms)。

数列中的每一项都和它的序号有关,排在第一位的数称为这个数列的第一项(首项),第n位的数称为这个数列的第n项。数列的一般形式写成 a1,a2,a3 … an … ,简记为 { an }

  • 1,0,1,0,1,0 ……
  • 0,1,2,3,4,5 ……
  • 1,2,4,8,16,32 ……

 

2.   通项公式

如果数列的第n 项与序号n之间可以可以用一个公式来表示,那么这个公式就叫做这个数列的通项公式

  • { an }: 2,0,2,0,2,0 ……    an = 1 + (–1) n+1
  • { an }: 0,1,2,3,4,5 ……    an = n – 1
  • { an }: 1,2,4,8,16,32 ……  an = 2 n–1


 

3.   等差数列(Arithmetic Sequence

一个数列从第二项起,每一项与它的前一项的等于一个常数,此数列为等差数列。这个常数叫做等差数列的公差(Common Difference),用字母d表示。通项公式: an = a1 + (n – 1) d

  • { an }: 0,1,2,3,4,5 ……           d = 1, an = n – 1
  • { an }: 1,5,9,13,17,21 ……    d = 4, an = 1 + (n – 1) × 4 = 4n – 3
  • { an }: 100,90,80,70,60 ……    d = –10, an = 100 + (n – 1) × (–10) = –10n + 110

若3个数a,A,b 成等差数列, A 是a与b的等差中项(Arithemetic mean)。

 

 

4.   等差数列的求和公式:

Sn = a1 + a2 + a3 + …… + an

Sn = a1 + (a1 + d) + (a1 + 2d) + …… + [ a1 + (n – 1)d ]              (1)

Sn = an + (an – d) + (an – 2d) + …… + [ an – (n – 1)d ]              (2)

(1) + (2)   = 2Sn = n (a1 + an)

∴                                          Sn = n (a1 + an) / 2

∵                                           an = a1 + (n – 1)d

∴                                          Sn = na1 + n (n – 1) d /2

 

 

5.   等比数列(Geometric Sequence

一个数列从第二项起,每一项与它的前一项的等于一个常数,此数列为等比数列。这个常数叫做等比数列的公比(Common Ratio),用字母q (q ≠0) 表示。通项公式:an = a1q n – 1

  • { an }: 1,2,4,8,16,32 ……     q = 2, an = 2 n–1
  • { an }: 243,81,27,9,3,1 ……   q = 1/3, an = 243 (1/3) n–1
  • { an }: 4,–8,16,–32,64 … …     q = –2, an = 4 (–2) n–1

若3个数a,G,b 成等差数列, G 是a与b的等比中项(Geometric mean)。

 

 

6.   等比数列的求和公式:

Sn = a1 + a2 + a3 + …… + an

Sn = a1 + a1q + a1q2 + …… + a1q n – 1              (1)

qSn = a1q + a1q2 + …… + a1q n – 1 + a1q n           (2)

(1) – (2)   (1 – q) Sn = a1 (1 – qn )

geometric_sequence

 

 

练习:

1. 如果a1, a2, … a8 为各项都大于零的等差数列,公差d ≠ 0,则()
A. a1a8 > a4a5
B. a1a8 = a4a5
C. a1a8 < a4a5
D. a1 + a8 > a4 + a5

 
2. 设等比数列 {an} 的前n项和为 Sn,若S6 / S3 = 3,则 S9 / S6 = ()
A. 2
B. 7/3
C. 8/3
D. 3

 
3. 在等差数列 {an} 中,满足 3a4 = 7a7,且a1 > 0,Sn 是数列 {an} 的前n项和,当 Sn 取的最大值时,n =    .

 
4. 根据数列的前几项,写出下列各数列的通项公式:
(1) 6, 66, 666, …
(2) 1, 3, 6, 10, 15, …
(3) 4/5, 1/2, 4/11, 2/7, …

 
5. 数列 {an} 的前n项和为 Sn.
(1) 若Sn = (-1)n+1·n,求 a5 + a6 及 an
(2) 若Sn = 3n + 2n + 1,求 an.

 

 

English:

数列 sequence, 等差数列 arithmetic sequence, 等比数列 Geometric Sequence

 

SAT:

The first term in a geometric sequence is 2, and the common radio is 3. The first term in an arithmetic sequence is 3, and the common difference is 3. Let set X be the set containing the first six terms of the geometric sequence and set Y be the set containing the first six terms of the arithmetic sequence. What is the sum of the elements in X ∩ Y?

 

 

数列,等差、等比数列求和公式