[用户互动]高中数学所有知识点

楼主^#

更多发布于：2019-02-24 08:07

高中数学第三章数列

考试内容：
©数列．
©等差数列及其通项公式．等差数列前n项和公式．
©等比数列及其通项公式．等比数列前n项和公式．
©考试要求：
©（1）理解数列的概念，了解数列通项公式的意义了解递推公式是给出数列的一种方法，并能根据递推公式写出数列的前几项．
©（2）理解等差数列的概念，掌握等差数列的通项公式与前n项和公式，并能解决简单的实际问题．
©（3）理解等比数列的概念，掌握等比数列的通项公式与前n项和公式，井能解决简单的实际问题．
§03. 数列知识要点

	等差数列	等比数列
定义
递推公式	；	；
通项公式		（）
中项	（）	（）
前项和
重要性质

1. ⑴等差、等比数列：

		等差数列	等比数列
定义
通项公式		=+（n-1）d=+（n-k）d=+-d
求和公式		[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image056.gif[/img]	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image058.gif[/img]
中项公式		A=[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image060.gif[/img] 推广：2[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image045.gif[/img]=[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image062.gif[/img]	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image064.gif[/img]。推广：[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image066.gif[/img]
性质	1	若m+n=p+q则 [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image068.gif[/img]	若m+n=p+q，则[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image070.gif[/img]。
	2	若[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image072.gif[/img]成A.P（其中[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image074.gif[/img]）则[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image076.gif[/img]也为A.P。	若[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image072.gif[/img]成等比数列（其中[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image074.gif[/img]），则[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image076.gif[/img]成等比数列。
	3	．[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image078.gif[/img] 成等差数列。	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image078.gif[/img]成等比数列。
	4	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image080.gif[/img]	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image082.gif[/img] ， [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image084.gif[/img] [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image086.gif[/img]
	5

⑵看数列是不是等差数列有以下三种方法：
①[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image088.gif[/img]
②2[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image090.gif[/img]([img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image092.gif[/img])
③[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image094.gif[/img]([img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image096.gif[/img]为常数).
⑶看数列是不是等比数列有以下四种方法：
①[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image098.gif[/img]
②[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image100.gif[/img]([img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image092.gif[/img]，[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image102.gif[/img])^①
注①：i. [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image104.gif[/img]，是a、b、c成等比的双非条件，即[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image104.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image106.gif[/img]a、b、c等比数列.
ii. [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image104.gif[/img]（ac＞0）→为a、b、c等比数列的充分不必要.
iii. [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image108.gif[/img]→为a、b、c等比数列的必要不充分.
iv. [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image110.gif[/img]且[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image112.gif[/img]→为a、b、c等比数列的充要.
注意：任意两数a、c不一定有等比中项，除非有ac＞0，则等比中项一定有两个.
③[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image114.gif[/img]([img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image116.gif[/img]为非零常数).
④正数列{[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image118.gif[/img]}成等比的充要条件是数列{[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image120.gif[/img]}（[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image122.gif[/img]）成等比数列.
⑷数列{[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image118.gif[/img]}的前[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image029.gif[/img]项和[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image124.gif[/img]与通项[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image118.gif[/img]的关系：[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image126.gif[/img]
[注]： ①[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image128.gif[/img]（[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image130.gif[/img]可为零也可不为零→为等差数列充要条件（即常数列也是等差数列）→若[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image130.gif[/img]不为0，则是等差数列充分条件）.
②等差{[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image132.gif[/img]}前n项和[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image134.gif[/img] →[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image136.gif[/img]可以为零也可不为零→为等差的充要条件→若[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image130.gif[/img]为零，则是等差数列的充分条件；若[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image130.gif[/img]不为零，则是等差数列的充分条件.
③非零常数列既可为等比数列，也可为等差数列.（不是非零，即不可能有等比数列）
2. ①等差数列依次每k项的和仍成等差数列，其公差为原公差的k²倍[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image139.gif[/img]；
②若等差数列的项数为2[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image141.gif[/img]，则[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image143.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image145.gif[/img]；
③若等差数列的项数为[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image147.gif[/img]，则[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image149.gif[/img]，且[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image151.gif[/img]，[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image153.gif[/img]
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image155.gif[/img].

3. 常用公式：①1+2+3 …+n =[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image157.gif[/img]

②[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image159.gif[/img]
③[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image161.gif[/img]
[注]：熟悉常用通项：9，99，999，…[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image163.gif[/img]； 5，55，555，…[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image165.gif[/img].
4. 等比数列的前[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image029.gif[/img]项和公式的常见应用题：
⑴生产部门中有增长率的总产量问题. 例如，第一年产量为[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image167.gif[/img]，年增长率为[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image169.gif[/img]，则每年的产量成等比数列，公比为[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image171.gif[/img]. 其中第[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image029.gif[/img]年产量为[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image173.gif[/img]，且过[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image029.gif[/img]年后总产量为：
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image175.gif[/img]
⑵银行部门中按复利计算问题. 例如：一年中每月初到银行存[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image167.gif[/img]元，利息为[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image169.gif[/img]，每月利息按复利计算，则每月的[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image167.gif[/img]元过[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image029.gif[/img]个月后便成为[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image177.gif[/img]元. 因此，第二年年初可存款：
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image179.gif[/img]=[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image181.gif[/img].
⑶分期付款应用题：[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image167.gif[/img]为分期付款方式贷款为a元；m为m个月将款全部付清；[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image169.gif[/img]为年利率.
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image185.gif[/img]
5. 数列常见的几种形式：
⑴[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image187.gif[/img]（p、q为二阶常数）[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image189.gif[/img]用特证根方法求解.
具体步骤：①写出特征方程[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image191.gif[/img]（[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image193.gif[/img]对应[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image195.gif[/img]，x对应[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image197.gif[/img]），并设二根[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image199.gif[/img]②若[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image201.gif[/img]可设[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image203.gif[/img]，若[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image205.gif[/img]可设[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image207.gif[/img]；③由初始值[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image209.gif[/img]确定[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image211.gif[/img].
⑵[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image213.gif[/img]（P、r为常数）[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image189.gif[/img]用①转化等差，等比数列；②逐项选代；③消去常数n转化为[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image216.gif[/img]的形式，再用特征根方法求[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image132.gif[/img]；④[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image219.gif[/img]（公式法），[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image211.gif[/img]由[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image209.gif[/img]确定.
①转化等差，等比：[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image221.gif[/img].
②选代法：[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image223.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image225.gif[/img]
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image227.gif[/img].
③用特征方程求解：[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image229.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image197.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image232.gif[/img].
④由选代法推导结果：[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image234.gif[/img].
6. 几种常见的数列的思想方法：
⑴等差数列的前[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image029.gif[/img]项和为[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image124.gif[/img]，在[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image236.gif[/img]时，有最大值. 如何确定使[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image124.gif[/img]取最大值时的[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image029.gif[/img]值，有两种方法：
一是求使[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image238.gif[/img]，成立的[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image029.gif[/img]值；二是由[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image240.gif[/img]利用二次函数的性质求[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image029.gif[/img]的值.
⑵如果数列可以看作是一个等差数列与一个等比数列的对应项乘积，求此数列前[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image029.gif[/img]项和可依照等比数列前[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image029.gif[/img]项和的推倒导方法：错位相减求和. 例如：[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image242.gif[/img]
⑶两个等差数列的相同项亦组成一个新的等差数列，此等差数列的首项就是原两个数列的第一个相同项，公差是两个数列公差[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image244.gif[/img]的最小公倍数.

2. 判断和证明数列是等差（等比）数列常有三种方法：(1)定义法:对于n≥2的任意自然数,验证[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image246.gif[/img]为同一常数。(2)通项公式法。(3)中项公式法:验证[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image248.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image250.gif[/img]都成立。
3. 在等差数列｛[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image045.gif[/img]｝中,有关S_n 的最值问题：(1)当[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image047.gif[/img]>0,d<0时，满足[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image253.gif[/img]的项数m使得[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image255.gif[/img]取最大值. (2)当[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image047.gif[/img]<0,d>0时，满足[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image257.gif[/img]的项数m使得[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image255.gif[/img]取最小值。在解含绝对值的数列最值问题时,注意转化思想的应用。
（三）、数列求和的常用方法
1. 公式法:适用于等差、等比数列或可转化为等差、等比数列的数列。

2.裂项相消法:适用于[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image259.gif[/img]其中{ [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image045.gif[/img]}是各项不为0的等差数列，c为常数；部分无理数列、含阶乘的数列等。
3.错位相减法:适用于[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image261.gif[/img]其中{ [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image045.gif[/img]}是等差数列，[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image263.gif[/img]是各项不为0的等比数列。

4.倒序相加法: 类似于等差数列前n项和公式的推导方法.
5.常用结论
1）: 1+2+3+...+n = [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image265.gif[/img]
2） 1+3+5+...+(2n-1) =[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image267.gif[/img]
3）[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image269.gif[/img]

4） [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image271.gif[/img]
5） [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image273.gif[/img] [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image275.gif[/img]
6） [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image277.gif[/img]

喜欢0

一个愿意陪你到老的人！