高考数学复习第六章数列与数学归纳法6.3等比数列及其前n项和市赛课公开课一等奖省名师优质课获奖.pptx

,6.3,等比数列及其前,n,项和,1/54,基础知识自主学习,课时训练,题型分类深度剖析,内容索引,2/54,基础知识自主学习,3/54,1.,等比数列定义,普通地，假如一个数列,，那么这个数列叫做等比数列，这个常数叫做等比数列,，通惯用字母,表示,(,q,0).,2.,等比数列通项公式,设等比数列,a,n,首项为,a,1,，公比为,q,，则它通项,a,n,.,3.,等比中项,假如在,a,与,b,中间插入一个数,G,，使,a,，,G,，,b,成等比数列，那么,G,叫做,a,与,b,.,知识梳理,从第,2,项起，每一项与它前一项比等于同一,常数,公比,q,a,1,q,n,1,等比中项,4/54,4.,等比数列惯用性质,(1),通项公式推广：,a,n,a,m,(,n,，,m,N,*,).,(2),若,a,n,为等比数列，且,k,l,m,n,(,k,，,l,，,m,，,n,N,*,),，则,_,.,5.,等比数列前,n,项和公式,等比数列,a,n,公比为,q,(,q,0),，其前,n,项和为,S,n,，,当,q,1,时，,S,n,na,1,；,q,n,m,a,k,a,l,a,m,a,n,5/54,6.,等比数列前,n,项和性质,公比不为,1,等比数列,a,n,前,n,项和为,S,n,，则,S,n,，,S,2,n,S,n,，,S,3,n,S,2,n,仍成等比数列，其公比为,.,q,n,6/54,等比数列,a,n,单调性,知识拓展,7/54,判断以下结论是否正确,(,请在括号中打,“”,或,“”,),(1),满足,a,n,1,qa,n,(,n,N,*,，,q,为常数,),数列,a,n,为等比数列,.(,),(2),G,为,a,，,b,等比中项,G,2,ab,.(,),(3),假如数列,a,n,为等比数列，,b,n,a,2,n,1,a,2,n,，则数列,b,n,也是等比数列,.(,),(4),假如数列,a,n,为等比数列，则数列,ln,a,n,是等差数列,.(,),思索辨析,8/54,考点自测,答案,解析,9/54,答案,解析,2.,设等比数列,a,n,前,n,项和为,S,n,，若,S,2,3,，,S,4,15,，则,S,6,等于,A.31 B.32 C.63 D.64,依据题意知，等比数列,a,n,公比不是,1.,由等比数列性质，得,(,S,4,S,2,),2,S,2,(,S,6,S,4,),，,即,12,2,3,(,S,6,15),，解得,S,6,63.,故选,C.,10/54,3.(,教材改编,),在,9,与,243,中间插入两个数，使它们同这两个数成等比数列，则,插入,两个数,分别,为,_.,设该数列公比为,q,，由题意知，,243,9,q,3,，,q,3,27,，,q,3.,插入两个数分别为,9,3,27,，,27,3,81.,答案,解析,27,，,81,11/54,答案,解析,11,设等比数列,a,n,公比为,q,，,8,a,2,a,5,0,，,8,a,1,q,a,1,q,4,0.,q,3,8,0,，,q,2,，,12/54,题型分类深度剖析,13/54,题型一等比数列基本量运算,答案,解析,解得,a,4,2.,设等比数列,a,n,公比为,q,，,14/54,(2),在各项均为正数等比数列,a,n,中，,a,2,，,a,4,2,，,a,5,成等差数列，,a,1,2,，,S,n,是数列,a,n,前,n,项和，则,S,10,S,4,等于,A.1 008 B.2 016,C.2 032 D.4 032,答案,解析,由题意知,2(,a,4,2),a,2,a,5,，,即,2(2,q,3,2),2,q,2,q,4,q,(2,q,3,2),，得,q,2,，,所以,S,10,S,4,2 016.,故选,B.,15/54,等比数列基本量运算是等比数列中一类基本问题，数列中有五个量,a,1,，,n,，,q,，,a,n,，,S,n,，普通能够,“,知三求二,”,，经过列方程,(,组,),可迎刃而解,.,思维升华,16/54,跟踪训练,1,(1)(,诸暨市质检,),已知等比数列,a,n,首项,a,1,1,，且,a,2,，,a,4,，,a,3,成等差数列，则数列,a,n,公比,q,_,，数列,a,n,前,4,项和,S,4,_.,由,a,2,，,a,4,，,a,3,成等差数列得,2,a,1,q,3,a,1,q,a,1,q,2,，,当,q,1,时，,S,4,4,a,1,4,，,答案,解析,17/54,(2)(,湖南,),设,S,n,为等比数列,a,n,前,n,项和，若,a,1,1,，且,3,S,1,，,2,S,2,，,S,3,成等差数列，则,a,n,_.,3,n,1,答案,解析,由,3,S,1,，,2,S,2,，,S,3,成等差数列知，,4,S,2,3,S,1,S,3,，,可得,a,3,3,a,2,，所以公比,q,3,，,故等比数列通项,a,n,a,1,q,n,1,3,n,1,.,18/54,题型二等比数列判定与证实,例,2,设数列,a,n,前,n,项和为,S,n,，已知,a,1,1,，,S,n,1,4,a,n,2.,(1),设,b,n,a,n,1,2,a,n,，证实：数列,b,n,是等比数列；,证实,19/54,由,a,1,1,及,S,n,1,4,a,n,2,，,得,a,1,a,2,S,2,4,a,1,2.,a,2,5,，,b,1,a,2,2,a,1,3.,由,，得,a,n,1,4,a,n,4,a,n,1,(,n,2),，,a,n,1,2,a,n,2(,a,n,2,a,n,1,)(,n,2).,b,n,a,n,1,2,a,n,，,b,n,2,b,n,1,(,n,2),，,故,b,n,是首项,b,1,3,，公比为,2,等比数列,.,20/54,(2),求数列,a,n,通项公式,.,解答,由,(1),知,b,n,a,n,1,2,a,n,32,n,1,，,故,a,n,(3,n,1)2,n,2,.,21/54,引申探究,若将例,2,中,“,S,n,1,4,a,n,2,”,改为,“,S,n,1,2,S,n,(,n,1),”,，其它不变，求数列,a,n,通项公式,.,由已知得,n,2,时，,S,n,2,S,n,1,n,.,S,n,1,S,n,2,S,n,2,S,n,1,1,，,a,n,1,2,a,n,1,，,a,n,1,1,2(,a,n,1),，,n,2,，,又,a,1,1,，,S,2,a,1,a,2,2,a,1,2,，,a,2,3,，,当,n,1,时上式也成立，,故,a,n,1,是以,2,为首项，以,2,为公比等比数列，,a,n,1,22,n,1,2,n,，,a,n,2,n,1.,解答,22/54,(1),证实一个数列为等比数列惯用定义法与等比中项法，其它方法只用于选择题、填空题中判定；若证实某数列不是等比数列，则只要证实存在连续三项不成等比数列即可,.,(2),利用递推关系时要注意对,n,1,时情况进行验证,.,思维升华,23/54,跟踪训练,2,已知数列,a,n,满足,a,1,1,，,a,n,1,3,a,n,1.,证实,24/54,证实,因为当,n,1,时，,3,n,1,2,3,n,1,，,25/54,题型三等比数列性质应用,例,3,(1),若等比数列,a,n,各项均为正数，且,a,10,a,11,a,9,a,12,2e,5,，则,ln,a,1,ln,a,2,ln,a,20,_.,答案,解析,50,因为,a,10,a,11,a,9,a,12,2,a,10,a,11,2e,5,，,所以,a,10,a,11,e,5,.,所以,ln,a,1,ln,a,2,ln,a,20,ln(,a,1,a,2,a,20,),ln,(,a,1,a,20,)(,a,2,a,19,),(,a,10,a,11,),ln(,a,10,a,11,),10,10ln(,a,10,a,11,),10ln e,5,50ln e,50.,26/54,方法一,S,6,S,3,1,2,，,a,n,公比,q,1.,S,3,，,S,6,S,3,，,S,9,S,6,也成等比数列，即,(,S,6,S,3,),2,S,3,(,S,9,S,6,),，,答案,解析,27/54,等比数列常见性质应用,等比数列性质应用能够分为三类：,(1),通项公式变形；,(2),等比中项变形；,(3),前,n,项和公式变形,.,依据题目条件，认真分析，发觉详细改变特征即可找出处理问题突破口,.,思维升华,28/54,跟踪训练,3,(1),已知在等比数列,a,n,中，,a,1,a,4,10,，则数列,lg,a,n,前,4,项和等于,A.4 B.3 C.2 D.1,答案,解析,前,4,项和,S,4,lg,a,1,lg,a,2,lg,a,3,lg,a,4,lg(,a,1,a,2,a,3,a,4,),，,又,等比数列,a,n,中，,a,2,a,3,a,1,a,4,10,，,S,4,lg 100,2.,29/54,(2),设等比数列,a,n,中，前,n,项和为,S,n,，已知,S,3,8,，,S,6,7,，则,a,7,a,8,a,9,等于,答案,解析,因为,a,7,a,8,a,9,S,9,S,6,，且公比不等于,1,，,在等比数列中，,S,3,，,S,6,S,3,，,S,9,S,6,也成等比数列，,即,8,，,1,，,S,9,S,6,成等比数列，,30/54,分类讨论思想在等比数列中应用,思想与方法系列,14,(1),利用等差数列性质求出等比数列公比，写出通项公式；,(2),求出前,n,项和，依据函数单调性证实,.,规范解答,思想方法指导,31/54,(1),解,设等比数列,a,n,公比为,q,，,因为,2,S,2,，,S,3,，,4,S,4,成等差数列，,所以,S,3,2,S,2,4,S,4,S,3,，即,S,4,S,3,S,2,S,4,，,32/54,33/54,34/54,课时训练,35/54,答案,解析,1,2,3,4,5,6,7,8,9,10,11,12,13,36/54,2.(,珠海模拟,),在等比数列,a,n,中，若,a,1,0,，,由,a,2,，,a,4,2,，,a,5,成等差数列，得,a,2,a,5,2(,a,4,2),，,即,2,q,2,q,4,2(2,q,3,2),，,(,q,2)(1,q,3,),0,，,解得,q,2,或,q,1(,舍去,),，,1,2,3,4,5,6,7,8,9,10,11,12,13,39/54,5.,已知数列,a,n,满足,log,3,a,n,1,log,3,a,n,1,(,n,N,*,),，且,a,2,a,4,a,6,9,，则,值是,答案,解析,由,log,3,a,n,1,log,3,a,n,1,(,n,N,*,),，,因为,a,5,a,7,a,9,(,a,2,a,4,a,6,),q,3,，,所以,a,5,a,7,a,9,9,3,3,3,5,.,所以,5.,1,2,3,4,5,6,7,8,9,10,11,12,13,40/54,6.(,铜仁质量检测,),在由正数组成等比数列,a,n,中，若,a,3,a,4,a,5,3,，则,sin(log,3,a,1,log,3,a,2,log,3,a,7,),值为,答案,解析,1,2,3,4,5,6,7,8,9,10,11,12,13,41/54,7.,设,S,n,为等比数列,a,n,前,n,项和，已知,3,S,3,a,4,2,，,3,S,2,a,3,2,，则公比,q,_.,答案,解析,4,由,，得,3,a,3,a,4,a,3,，即,4,a,3,a,4,，,1,2,3,4,5,6,7,8,9,10,11,12,13,42/54,8.,设各项都是正数等比数列,a,n,，,S,n,为前,n,项和且,S,10,10,，,S,30,70,，那么,S,40,_.,答案,解析,150,依题意，知数列,a,n,公比,q,1,，数列,S,10,，,S,20,S,10,，,S,30,S,20,，,S,40,S,30,成等比数列，,所以有,(,S,20,S,10,),2,S,10,(,S,30,S,20,),，,即,(,S,20,10),2,10(70,S,20,),，故,S,20,20,或,S,20,30,；,又,S,20,0,，所以,S,20,30,，,S,20,S,10,20,，,S,30,S,20,40,，,故,S,40,S,30,80,，,S,40,150.,1,2,3,4,5,6,7,8,9,10,11,12,13,43/54,9.,已知数列,a,n,前,n,项和为,S,n,，且满足,a,n,S,n,1(,n,N,*,),，则通项,a,n,_.,答案,解析,a,n,S,n,1,，,1,2,3,4,5,6,7,8,9,10,11,12,13,44/54,1 024,a,4,b,1,b,2,b,3,，,，,a,n,b,1,b,2,b,3,b,n,1,，,a,21,b,1,b,2,b,3,b,20,(,b,10,b,11,),10,2,10,1 024.,答案,解析,1,2,3,4,5,6,7,8,9,10,11,12,13,45/54,11.,已知,a,n,是等差数列，满足,a,1,3,，,a,4,12,，数列,b,n,满足,b,1,4,，,b,4,20,，且,b,n,a,n,是等比数列,.,(1),求数列,a,n,和,b,n,通项公式；,解答,1,2,3,4,5,6,7,8,9,10,11,12,13,46/54,设等差数列公差为,d,，,所以,a,n,a,1,(,n,1),d,3,n,(,n,N,*,).,设等比数列,b,n,a,n,公比为,q,，,所以,b,n,a,n,(,b,1,a,1,),q,n,1,2,n,1,.,从而,b,n,3,n,2,n,1,(,n,N,*,).,1,2,3,4,5,6,7,8,9,10,11,12,13,47/54,(2),求数列,b,n,前,n,项和,.,解答,由,(1),知,b,n,3,n,2,n,1,(,n,N,*,),，,1,2,3,4,5,6,7,8,9,10,11,12,13,48/54,解答,1,2,3,4,5,6,7,8,9,10,11,12,13,49/54,(2),求,a,n,通项公式,.,解答,1,2,3,4,5,6,7,8,9,10,11,12,13,50/54,解答,1,2,3,4,5,6,7,8,9,10,11,12,13,51/54,b,n,a,2,n,a,2,n,1,，,1,2,3,4,5,6,7,8,9,10,11,12,13,52/54,1,2,3,4,5,6,7,8,9,10,11,12,13,53/54,(2),求,T,2,n,.,解答,1,2,3,4,5,6,7,8,9,10,11,12,13,54/54,