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13636.com:等差数列的概念与性质

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等差数列的概念与性质

主讲人:尹连辉 2013.6.20

基础知识回顾
一. 二. 三.

等差数列的概念; 等差数列的通项公式及前n项和公式; 等差数列的性质;

2

基础知识回顾
(一)等差数列的概念
如果一个数列从第二项起每一项减去它的前 一项所得的差都是同一个常数,那么这个数 列就叫做等差数列

an ?1 ? an ? d (n ? N ? ) 或者(an ? an -1 ? d (n ? 2, n ? N ? )

典例导悟

例1.已知数列 ?an ? 中,1 a

一、等差数列的判定与证明
? 2, a2 ? 3, 其前n项和

sn 满足 an?1 ? sn?1 ? sn ? 1

(n ? 2, n ? N ? ) 求证数列 ?an ? 为等差数列,并求 ?an ? 的通项公式 。

例2.已知数列?an ? 满足 a1 ? 4,

an ? 4 ?

(1)求证:数列?bn ?是等差数列; (2)求数列 ?an ?的通项公式。

4 1 (n ? 1),记 bn ? . an ?1 an ? 2

基础知识回顾
二、等差数列的通项公式及前n项和公式

an ? a1 ? (n ?1)d

推导方法:累加法

推广的通项公式:an ? am +(n ? m)d

a1 ? an Sn ? ?n 2 n(n ? 1) ? na1 ? d 2

an ? pn ? (p, q为常数) q
推导方法:倒序相加法

典例导悟

二、等差数列基本量的运算
1 通项公式中
例3.在等差数列 ?an ? 中, (1)已知 a1 ? 6, d ? 3, 求 (2)已知

a; 8

a2 ? 12, an ? ?20,d ? ?2 ,求n; 1 a7 ? ? , d ? ?2, a (3)已知 求 1 2 (4)已知 a4 ? 10, a10 ? 4, 求 a7 和d;
思考: 在等差数列?an ?中,已知a p ? q, aq ? p, 且 p ? q, 求ap ? q

典例导悟

思考: 已知等差数列{an}的前n项和为Sn,若 m

s ? n, sn ? m, 且 m ? n, 求s m+n

典例导悟

变式 : 若数列{an }为等差数列,且S p ? Sq , ( p ? q, p, q ? N ? ) 求S p ? q

p( p ? 1) q(q ? 1) 解 : ? S p ? S q ? pa1 ? d ? qa1 ? d 2 2 ( p ? q)( p ? q ? 1) ? ( p ? q)a1 ? d ?0 2 p ? q ?1 ? p ? q ? a1 ? d ?0 2 思考:试用图象法验证? p ? q ?1 ? S p ? q ? ( p ? q)(a1 ? d) ? 0 2

评注: 恰当运用前n项和公式以及整体代换是解决本题的关键。

基础知识回顾
三、等差数列的性质
(1)等差中项的概念:
a?b 如果a、A、b这三个数成等差数列,那么 A ? 2

叫做a和b的等差中项。

基础知识回顾
2.对称性: 在等差数列?an ?中, 若m ? n ? p ? q, m, n, p, q ? N ? 则am ? an ? a p ? aq , 特别地:若m ? n ? 2 p则am ? an ? 2a p
思考:若p ? q ? n ? 3m, 则a p ? aq ? an ? 3am成立吗? 正确

典例导悟

(四)等差数列对称性的应用

例6.在等差数列{an }中,
(1)若 a3 ? a4 ? a5 ? a6 ? a7 ? 450 ,求 (2)若 a3 ? a5 ? a7 ? a9 ? a11 ? 100, 求

a2 ? a8 3a9 -a13

(3)若a2 ? 4, a6 ? 12, 求a7 ? a8 ? ? ? a21的值.
变式1:等差数列?an ? 的前n项和为 sn ,已知 am ?1 ? am ?1 ? am
2

? 0,

s2 m ?1 ? 38,

则m=————

变式2:若等差数列共有n项,且前四项之和为21,后四项之和为67,前n项和 为Sn=286,求n.

3.两个等差数列前n项和之比问 题

典例导悟

(四)等差数列公式的应用

变式 : 设等差数列{a n }的前n项和为S n , S9 9 若a5 ? 5a3 , 则 ? ______ S5

4.片段和: 在等差数列?an ?中,S n是其前n项的和,k ? N ? , 那么, S k , S 2 k ? S k , S3k ? S 2 k 成等差数列。如何证明? a1 ? ak 略证:Sk ? ?k (1) 2 ak ?1 ? a2 k S2k ? S k ? ak ?1 ? ak ? 2 ? ? ? a2 k ? ? k (2) 2 a2 k ?1 ? a3k S3k ? S 2 k ? ?k (3) 2 k
2

(1) ? (3)得:Sk ? ( S 3k ? S 2 k ) ? ?

?a1 ? ak ? a2 k ?1 ? a3k ?

k ??a1 ? a2 k ?1 ? ? ?ak ? a3k ?? ? ?ak ?1 ? a2 k ?k ? 2?S 2 k ? S k ? 2

? Sk , S2k ? Sk , S3k ? S2k 成等差数列

典例导悟

(四)等差数列公式的应用

[解]

S4 1 ∵S =3,故设S4=x,则S8=3x. 8

由于S4,S8-S4,S12-S8,S16-S12成等差数列,且S4= x,S8-S4=3x-x=2x, ∴新数列公差为x. ∴S12-S8=3x,S16-S12=4x, ∴S12=3x+S8=3x+3x=6x,而S16=S12+4x=6x+4x =10x. S8 3x 3 ∴ = = . S16 10x 10

变式训练1 (1)等差数列{an}中,a1+a2+a3+a4= 30,a5+a6+a7+a8=80,则a9+a10+a11+a12=________. (2)一个等差数列前n项和为25,前2n项和为100,求其 前3n项的和.

小结:
本节课复习了等差数列的基础知识, 1.要从公式的顺向、逆向、变式多角度 的掌握等差数列的基本公式。 2.要理解等差数列的定义,掌握证明等 差数列的方法;熟练运用通项公式和求 和公式,注意运用方程的思想、函数的 思想及数形结合思想分析和解决问题。


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