高考数学复习第六章数列6.4数列求和市赛课公开课一等奖省名师优质课获奖课件.pptx

剖析题型 提炼方法,实验解读,构建知识网络 强化答题语句,探究高考 明确考向,6.4,数列求和,第六章数列,1/86,基础知识自主学习,课时作业,题型分类深度剖析,内容索引,2/86,基础知识自主学习,3/86,1.,等差数列前,n,项和公式,S,n,_,.,2.,等比数列前,n,项和公式,知识梳理,，,q,1.,4/86,3.,一些常见数列前,n,项和公式,(1)1,2,3,4,n,_,.,(2)1,3,5,7,2,n,1,.,(3)2,4,6,8,2,n,.,(4)1,2,2,2,n,2,.,n,2,n,(,n,1),5/86,数列求和惯用方法,(1),公式法,直接利用等差、等比数列求和公式求和,.,(2),分组转化法,把数列转化为几个等差、等比数列，再求解,.,【,知识拓展,】,6/86,(3),裂项相消法,把数列通项拆成两项之差求和，正负相消剩下首尾若干项,.,常见裂项公式,7/86,(4),倒序相加法,把数列分别正着写和倒着写再相加，即等差数列求和公式推导过程推广,.,(5),错位相减法,主要用于一个等差数列与一个等比数列对应项相乘所得数列求和,.,(6),并项求和法,一个数列前,n,项和中，可两两结合求解，则称之为并项求和,.,形如,a,n,(,1),n,f,(,n,),类型，可采取两项合并求解,.,8/86,题组一思索辨析,1.,判断以下结论是否正确,(,请在括号中打或,“”,),(1),假如数列,a,n,为等比数列，且公比不等于,1,，则其前,n,项和,S,n,.,(,),(2),当,n,2,时，,.(,),(3),求,S,n,a,2,a,2,3,a,3,na,n,之和时，只要把上式等号两边同时乘以,a,即可依据错位相减法求得,.(,),基础自测,1,2,3,4,5,6,9/86,(4),数列,前,n,项和为,n,2,.(,),(5),推导等差数列求和公式方法叫做倒序求和法，利用此法可求得,sin,2,1,sin,2,2,sin,2,3,sin,2,88,sin,2,89,44.5.(,),(6),假如数列,a,n,是周期为,k,周期数列，那么,S,km,mS,k,(,m,，,k,为大于,1,正整数,).(,),1,2,3,4,5,6,10/86,题组二教材改编,2.,P61A,组,T5,一个球从,100 m,高处自由落下，每次着地后又跳回到原高度二分之一再落下，当它第,10,次着地时，经过旅程是,A.100,200(1,2,9,)B.100,100(1,2,9,),C.200(1,2,9,)D.100(1,2,9,),答案,解析,1,2,3,4,5,6,解析,第,10,次着地时，经过旅程为,100,2(50,25,100,2,9,),100,2,100,(2,1,2,2,2,9,),11/86,3.,P61A,组,T4(3),1,2x,3,x,2,nx,n,1,_(,x,0,且,x,1).,答案,1,2,3,4,5,6,解析,设,S,n,1,2,x,3,x,2,nx,n,1,，,则,xS,n,x,2,x,2,3,x,3,nx,n,，,得,(1,x,),S,n,1,x,x,2,x,n,1,nx,n,解析,12/86,题组三易错自纠,4.(,潍坊调研,),设,a,n,是公差不为,0,等差数列，,a,1,2,，且,a,1,，,a,3,，,a,6,成等比数列，则,a,n,前,n,项和,S,n,等于,答案,1,2,3,4,5,6,解析,13/86,1,2,3,4,5,6,解析,设等差数列公差为,d,，则,a,1,2,，,a,3,2,2,d,，,a,6,2,5,d,.,即,(2,2,d,),2,2(2,5,d,),，整理得,2,d,2,d,0.,14/86,5.(,日照质检,),数列,a,n,通项公式为,a,n,(,1),n,1,(4,n,3),，则它前,100,项之和,S,100,等于,A.200 B.,200,C.400 D.,400,解析,答案,1,2,3,4,5,6,解析,S,100,(41,3),(42,3),(43,3),(4100,3),4(1,2),(3,4),(99,100),4(,50),200.,15/86,6.,数列,a,n,通项公式为,a,n,，其前,n,项和为,S,n,，则,S,2 017,解析,答案,1,2,3,4,5,6,故,S,4,a,1,a,2,a,3,a,4,2.,a,5,0,，,a,6,6,，,a,7,0,，,a,8,8,，,故,a,5,a,6,a,7,a,8,2,，,周期,T,4.,1 008,1 008.,16/86,题型分类深度剖析,17/86,解答,题型一分组转化法求和,师生共研,典例,(,合肥质检,),已知数列,a,n,前,n,项和,S,n,，,n,N,*,.,(1),求数列,a,n,通项公式；,18/86,解,当,n,1,时，,a,1,S,1,1,；,a,1,也满足,a,n,n,，,故数列,a,n,通项公式为,a,n,n,.,19/86,解答,(2)设,b,n,2,a,n,(,1),n,a,n,，求数列,b,n,前,2,n,项和,.,解,由,(1),知,a,n,n,，故,b,n,2,n,(,1),n,n,.,记数列,b,n,前,2,n,项和为,T,2,n,，则,T,2,n,(2,1,2,2,2,2,n,),(,1,2,3,4,2,n,).,记,A,2,1,2,2,2,2,n,，,B,1,2,3,4,2,n,，,B,(,1,2),(,3,4),(2,n,1),2,n,n,.,故数列,b,n,前,2,n,项和,T,2,n,A,B,2,2,n,1,n,2.,20/86,本例,(2),中，求数列,b,n,前,n,项和,T,n,.,引申探究,解答21/86,解,由,(1),知,b,n,2,n,(,1),n,n,.,当,n,为偶数时，,T,n,(2,1,2,2,2,n,),1,2,3,4,(,n,1),n,当,n,为奇数时，,T,n,(2,1,2,2,2,n,),1,2,3,4,(,n,2),(,n,1),n,22/86,23/86,分组转化法求和常见类型,(1),若,a,n,b,n,c,n,，且,b,n,，,c,n,为等差或等比数列，可采取分组求和法求,a,n,前,n,项和,.,(2),通项公式为,a,n,数列，其中数列,b,n,，,c,n,是等比数,列或等差数列，可采取分组求和法求和,.,提醒：一些数列求和是将数列转化为若干个可求和新数列和或差，从而求得原数列和，注意在含有字母数列中对字母讨论,.,思维升华,24/86,跟踪训练,等差数列,a,n,前,n,项和为,S,n,，数列,b,n,是等比数列，满足,a,1,3,，,b,1,1,，,b,2,S,2,10,，,a,5,2,b,2,a,3,.,(1),求数列,a,n,和,b,n,通项公式；,解答,解,设数列,a,n,公差为,d,，数列,b,n,公比为,q,，,a,n,3,2(,n,1),2,n,1,，,b,n,2,n,1,.,25/86,解答,26/86,T,2,n,(,c,1,c,3,c,2,n,1,),(,c,2,c,4,c,2,n,),27/86,典例,(,天津,),已知,a,n,为等差数列，前,n,项和为,S,n,(,n,N,*,),，,b,n,是首项为,2,等比数列，且公比大于,0,，,b,2,b,3,12,，,b,3,a,4,2,a,1,，,S,11,11,b,4,.,(1),求,a,n,和,b,n,通项公式；,题型二错位相减法求和,师生共研,解答,28/86,解,设等差数列,a,n,公差为,d,，等比数列,b,n,公比为,q,.,由已知,b,2,b,3,12,，得,b,1,(,q,q,2,),12,，而,b,1,2,，,所以,q,2,q,6,0.,又因为,q,0,，解得,q,2,，所以,b,n,2,n,.,由,b,3,a,4,2,a,1,，可得3,d,a,1,8,，,由,S,11,11,b,4,，可得,a,1,5,d,16,，,联立,，解得,a,1,1,，,d,3,，由此可得,a,n,3,n,2.,所以数列,a,n,通项公式为,a,n,3,n,2,，数列,b,n,通项公式为,b,n,2,n,.,29/86,解答,(2),求数列,a,2,n,b,2,n,1,前,n,项和,(,n,N,*,).,30/86,解,设数列,a,2,n,b,2,n,1,前,n,项和为,T,n,，由,a,2,n,6,n,2,，,b,2,n,1,2,4,n,1,，得,a,2,n,b,2,n,1,(3,n,1),4,n,，,故,T,n,2,4,5,4,2,8,4,3,(3,n,1),4,n,，,4,T,n,2,4,2,5,4,3,8,4,4,(3,n,4),4,n,(3,n,1),4,n,1,，,，得,3,T,n,2,4,3,4,2,3,4,3,3,4,n,(3,n,1),4,n,1,(3,n,2),4,n,1,8,，,31/86,错位相减法求和时注意点,(1),要善于识别题目类型，尤其是等比数列公比为负数情形,.,(2),在写出,“,S,n,”,与,“,qS,n,”,表示式时应尤其注意将两式,“,错项对齐,”,方便下一步准确写出,“,S,n,qS,n,”,表示式,.,(3),在应用错位相减法求和时，若等比数列公比为参数，应分公比等于,1,和不等于,1两种情况求解,.,思维升华,32/86,跟踪训练,(,阜阳调研,),设等差数列,a,n,公差为,d,，前,n,项和为,S,n,，等比数列,b,n,公比为,q,，已知,b,1,a,1,，,b,2,2,，,q,d,，,S,10,100.,(1),求数列,a,n,，,b,n,通项公式；,解答,33/86,解答,34/86,可得,35/86,典例,(,郑州市第二次质量预测,),已知数列,a,n,前,n,项和为,S,n,，,a,1,2,，且满足,S,n,a,n,1,n,1(,n,N,*,).,(1),求数列,a,n,通项公式；,题型三裂项相消法求和,多维探究,解答,36/86,两式相减，并化简，得,an,1,3,a,n,2,，,即,a,n,1,1,3(,a,n,1),，,又,a,1,1,2,1,3,0,，,所以,a,n,1,是以,3,为首项，,3,为公比等比数列，,所以,a,n,1,(,3)3,n,1,3,n,.,故,a,n,3,n,1.,37/86,证实,38/86,证实,由,b,n,log,3,(,a,n,1),log,3,3,n,n,，,39/86,典例,已知函数,f,(,x,),x,图象过点,(4,2),，令,a,n,，,n,N,*,.,记数列,a,n,前,n,项和为,S,n,，则,S,2 017,_.,解析,答案,40/86,41/86,思维升华,42/86,跟踪训练,(,届贵州遵义航天高级中学模拟,),已知等差数列,a,n,满足,(,a,1,a,2,),(,a,2,a,3,),(,a,n,a,n,1,),2,n,(,n,1).,(1),求数列,a,n,通项公式；,解答,解,设等差数列,a,n,公差为,d,，,当,n,1,时，,a,1,a,2,4,，,当,n,2,时，,a,1,a,2,a,2,a,3,12,，即,4,a,2,12,，,a,2,3,，,a,1,1,，,d,a,2,a,1,2,，,等差数列,a,n,通项公式,a,n,1,2(,n,1),2,n,1,，,a,n,2,n,1.,43/86,解答,S,n,b,1,b,2,b,3,b,n,44/86,典例,(12分,),在数列,a,n,中，,a,1,2,，,a,n,1,a,n,(,n,N,*,).,(1),求数列,a,n,通项公式；,审题路线图,四审结构定方案,审题路线图,规范解答,45/86,审题路线图,46/86,规范解答,47/86,48/86,(3),证实,当,n,2,时，,49/86,课时作业,50/86,基础,保分练,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,答案,解析,51/86,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,52/86,2.(,长春调研,),数列,a,n,前,n,项和为,S,n,，已知,S,n,1,23,4,(,1),n,1,n,，则,S,17,等于,A.9 B.8,C.17 D.16,解析,答案,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,解析,S,17,1,2,3,4,5,6,15,16,17,1,(,2,3),(,4,5),(,6,7),(,14,15),(,16,17),1,1,1,1,9.,53/86,3.,在数列,a,n,中，若,a,n,1,(,1),n,a,n,2,n,1,，则数列,a,n,前,12,项和等于,A.76 B.78 C.80 D.82,解析,答案,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,解析,由已知,a,n,1,(,1),n,a,n,2,n,1,，,得,a,n,2,(,1),n,1,a,n,1,2,n,1,，,得,a,n,2,a,n,(,1),n,(2,n,1),(2,n,1),，,取,n,1,5,9,及,n,2,6,10,，,结果相加可得,S,12,a,1,a,2,a,3,a,4,a,11,a,12,78.,故选,B.,54/86,解析,答案,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,4.(,高安中学模拟,),已知数列,5,6,1,，,5,，,，该数列特点是从第二项起，每一项都等于它前后两项之和，则这个数列前,16,项之和,S,16,等于,A.5 B.6,C.7 D.16,55/86,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,解析,依据题意这个数列前,8,项分别为,5,6,1,，,5,，,6,，,1,5,6,，发觉从第,7,项起，数字重复出现，所以此数列为周期数列，且周期为,6,，前,6,项和为,5,6,1,(,5),(,6),(,1),0.,又因为,16,2,6,4,，所以这个数列前,16,项之和,S,16,2,0,7,7.,故选,C.,56/86,A.0 B.100 C.,100 D.10 200,解析,答案,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,解析,由题意，得,a,1,a,2,a,3,a,100,1,2,2,2,2,2,3,2,3,2,4,2,4,2,5,2,99,2,100,2,100,2,101,2,(1,2),(3,2),(4,3),(99,100),(101,100),(1,2,99,100),(2,3,100,101),50,101,50,103,100.,故选,B.,57/86,6.(,开封调研,),已知数列,a,n,满足,a,1,1,，,a,n,1,a,n,2,n,(,n,N,*,),，则,S,2 018,等于,A.2,2 018,1 B.3,2,1 009,3,C.3,2,1 009,1 D.3,2,1 008,2,解析,答案,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,58/86,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,a,1,，,a,3,，,a,5,，,成等比数列；,a,2,，,a,4,，,a,6,，,成等比数列，,S,2 018,a,1,a,2,a,3,a,4,a,5,a,6,a,2 017,a,2 018,(,a,1,a,3,a,5,a,2 017,),(,a,2,a,4,a,6,a,2 018,),59/86,解析,答案,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,7.(,全国,),等差数列,a,n,前,n,项和为,S,n,，,a,3,3,，,S,4,10,，则,_.,60/86,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,解析,设等差数列,a,n,公差为,d,，则,61/86,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,62/86,解析,答案,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,8.(,商丘质检,),有穷数列,1,1,2,1,2,4,，,，,1,2,4,2,n,1,全部项和为,_.,2,n,1,2,n,63/86,解析,答案,1,2,3,4,5,6,7,8,9,10,1112,13,14,15,16,120,64/86,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,S,n,a,1,a,2,a,n,65/86,10.(,安阳二模,),已知数列,a,n,中，,a,n,4,n,5,，等比数列,b,n,公比,q,满足,q,a,n,a,n,1,(,n,2),且,b,1,a,2,，则,|,b,1,|,|,b,2,|,|,b,3,|,|,b,n,|,_.,解析,答案,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,4,n,1,解析,由已知得,b,1,a,2,3,，,q,4,，,b,n,(,3),(,4),n,1b,n,|,3,4,n,1,，,即,|,b,n,|,是以,3,为首项，,4,为公比等比数列，,66/86,解答,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,(1),求数列,a,n,通项公式；,67/86,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,解,设数列,a,n,公差为,d,，,所以,a,1,a,2,3.,所以,a,2,a,3,15.,由,解得,a,1,1,，,d,2,，,所以,a,n,2,n,1.,经检验，符合题意,.,68/86,解答,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,(2),设,b,n,(,a,n,1),，求数列,b,n,前,n,项和,T,n,.,69/86,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,解,由,(1),知,b,n,2,n,2,2,n,1,n,4,n,，,所以,T,n,14,1,24,2,n,4,n,，,所以,4,T,n,14,2,24,3,n,4,n,1,，,两式相减，得,3,T,n,4,1,4,2,4,n,n,4,n,1,70/86,12.(,贵阳一模,),已知数列,a,n,前,n,项和是,S,n,，且,S,n,1(,n,N,*,).,(1),求数列,a,n,通项公式；,解答,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,71/86,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,解,当,n,1,时，,a,1,S,1,，,72/86,解答,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,73/86,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,74/86,技能提升练,解析,答案,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,75/86,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,解析,由题意可得,a,n1,a,n,n,1,，,则,a,1,1,，,a,2,a,1,2,，,a,3,a,2,3,，,，,a,n,a,n,1,n,，,76/86,答案,解析,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,1 008,77/86,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,78/86,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,79/86,拓展冲刺练,答案,解析,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,80/86,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,解得,a,1,3,或,a,1,0.,由,a,n,0,，得,a,1,3.,所以,(,a,n,1,a,n,)(,a,n,1,a,n,3),0.,因为,a,n,0,，所以,a,n,1,a,n,0,，,a,n,1,a,n,3.,即数列,a,n,是以,3,为首项，,3,为公差等差数列，,所以,a,n,3,3(,n,1),3,n,.,81/86,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,所以,b,n,82/86,答案,解析,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,9,83/86,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,又,a,n,为正项数列，,a,n,1,a,n,1,0,，,即,a,n,1,a,n,1.,数列,a,n,是以,1,为首项，,1,为公差等差数列,.,a,n,n,，,84/86,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,85/86,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,令,n,1,t,2,，,1,n,100,，,n,3,8,15,24,35,48,63,80,99,，共,9,个数,.,T,1,，,T,2,，,T,3,，,，,T,100,中有理数个数为,9.,86/86,