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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
1. ⑴等差、等比数列:
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Sn 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] |
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沙发#
发布于:2019-02-24 08:08
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