高考数学 第五章第三节 等比数列及其前n项和课件 新人教A版 课件.ppt

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,*,*,1,(2010,重庆高考,),在等比数列,a,n,中，,a,2010,8,a,2007,，则,公比,q,的值为,(,),A,2,B,3,C,4 D,8,答案：,A,2,等比数列,a,n,中,a,5,4,，则,a,2,a,8,等于,(,),A,4 B,8,C,16 D,32,答案：,C,答案：,D,4,已知等比数列,a,n,各项都是正数，,a,1,3,，,a,1,a,2,a,3,21,，则,a,3,a,4,a,5,_.,解析：,a,1,a,2,a,3,21,，,a,1,(1,q,q,2,),21,又,a,1,3,，,1,q,q,2,7,解之得,q,2,或,q,3(,舍,),a,3,a,4,a,5,q,2,(,a,1,a,2,a,3,),421,84.,答案：,84,5,在数列,a,n,，,b,n,中，,b,n,是,a,n,与,a,n,1,的等差中项，,a,1,2,，,且对任意,n,N,*,，都有,3,a,n,1,a,n,0,，则,b,n,的通项公式,b,n,_.,1,等比数列的相关概念,a,1,q,n,1,相关名词,等比数列,a,n,的有关概念及公式,前,n,项,和公式,等比中项,设,a,、,b,为任意两个同号的实数，则,a,、,b,的等比中项,G,a,m,a,n,a,p,a,q,S,m,(,S,3,m,S,2,m,),已知数列,a,n,的首项,a,1,5,，前,n,项和为,S,n,，且,S,n,1,2,S,n,n,5,，,n,N,*,.,(1),证明：数列,a,n,1,是等比数列；,(2),求,a,n,的通项公式以及,S,n,.,考点一,等比数列的判定与证明,自主解答,(1),证明：由已知,S,n,1,2,S,n,n,5,，,n,N,*,，,可得,n,2,时，,S,n,2,S,n,1,n,4,，,两式相减得,S,n,1,S,n,2(,S,n,S,n,1,),1,，,即,a,n,1,2,a,n,1,，从而,a,n,1,1,2(,a,n,1),，,设数列,a,n,的前,n,项和为,S,n,，已知,a,1,2,a,2,3,a,3,na,n,(,n,1),S,n,2,n,(,n,N,*,),(1),求,a,2,，,a,3,的值；,(2),求证数列,S,n,2,是等比数列,解：,(1),a,1,2,a,2,3,a,3,na,n,(,n,1),S,n,2,n,(,n,N,*,),，,当,n,1,时，,a,1,21,2,，,当,n,2,时，,a,1,2,a,2,(,a,1,a,2,),4,，,a,2,4,，,当,n,3,时，,a,1,2,a,2,3,a,3,2(,a,1,a,2,a,3,),6,，,a,3,8.,(2),证明：,a,1,2,a,2,3,a,3,na,n,(,n,1),S,n,2,n,(,n,N,*,),，,当,n,2,时，,a,1,2,a,2,3,a,3,(,n,1),a,n,1,(,n,2),S,n,1,2(,n,1),，,得，,na,n,(,n,1),S,n,(,n,2),S,n,1,2,n,(,S,n,S,n,1,),S,n,2,S,n,1,2,na,n,S,n,2,S,n,1,2,，,考点二,等比数列的基本运算,在等比数列,a,n,中，已知,a,6,a,4,24,，,a,3,a,5,64.,求,a,n,前,8,项的和,S,8,.,自主解答,设数列,a,n,的首项为,a,1,，公比为,q,，由已知条件得：,a,6,a,4,a,1,q,3,(,q,2,1),24.(*),a,3,a,5,(,a,1,q,3,),2,64.,a,1,q,3,8.,将,a,1,q,3,8,代入,(*),式，,得,q,2,2(,舍去,),，,已知正项等比数列,a,n,中，,a,1,a,5,2,a,2,a,6,a,3,a,7,100,，,a,2,a,4,2,a,3,a,5,a,4,a,6,36,，求数列,a,n,的通项,a,n,和前,n,项和,S,n,.,(1),在等比数列,a,n,中，若,a,1,a,2,a,3,a,4,1,，,a,13,a,14,a,15,a,16,8,，求,a,41,a,42,a,43,a,44,.,(2),有四个正数，前三个数成等差数列，其和为,48,，后,三个数成等比数列，其最后一个数为,25,，求此四个数,考点三,等比数列的性质及应用,法二：,由性质可知，依次,4,项的积为等比数列，,设公比为,q,，,T,1,a,1,a,2,a,3,a,4,1,，,T,4,a,13,a,14,a,15,a,16,8,，,T,4,T,1,q,3,1,q,3,8.,q,2.,T,11,a,41,a,42,a,43,a,44,T,1,q,10,2,10,1 024.,(2),设前三个数分别为,a,d,，,a,，,a,d,(,d,为公差,),，,由题意知，,(,a,d,),a,(,a,d,),48,，,解得,a,16.,又后三个数成等比数列，即,16,16,d,25,成等比数列，,(16,d,),2,1625.,解之得，,d,4,，或,d,36.,因四个数均为正数，故,d,36,应舍去，,所以所求四个数依次是,12,16,20,25.,将问题,(1),中,“,a,1,a,2,a,3,a,4,1,，,a,13,a,14,a,15,a,16,8,”,改为,“,a,1,a,2,a,3,7,，,a,1,a,2,a,3,8,”,，求,a,n,的通项公式,(1),已知等比数列,a,n,满足,a,n,0,，,n,1,2,，,，且,a,5,a,2,n,5,2,2,n,(,n,3),，则当,n,1,时，求,log,2,a,1,log,2,a,3,log,2,a,2,n,1,的值,(2),各项均为正数的等比数列,a,n,的前,n,项和为,S,n,，若,S,n,2,，,S,3,n,14,，求,S,4,n,的值,(2),由等比数列性质：,S,n,，,S,2,n,S,n,，,S,3,n,S,2,n,，,S,4,n,S,3,n,成等比数列，,则,(,S,2,n,S,n,),2,S,n,(,S,3,n,S,2,n,),，,(,S,2,n,2),2,2(14,S,2,n,),又,S,2,n,0,得,S,2,n,6,，又,(,S,3,n,S,2,n,),2,(,S,2,n,S,n,)(,S,4,n,S,3,n,),，,(14,6),2,(6,2)(,S,4,n,14),，解得,S,4,n,30.,考点四,等比数列的综合应用,自主解答,(1),S,n,1,3,S,n,2,，,S,n,1,1,3(,S,n,1),又,S,1,1,3,，,S,n,1,是首项为,3,，公比为,3,的等比数列且,S,n,3,n,1,，,n,N,*,.,(2),n,1,时，,a,1,S,1,2,，,n,1,时，,a,n,S,n,S,n,1,(3,n,1),(3,n,1,1),(2),由,(1),知,lg(1,a,n,),2,n,1,lg(1,a,1,),2,n,1,lg3,，,1,a,n,32,n,1,.(*),T,n,(1,a,1,)(1,a,2,)(1,a,n,),.,由,(*),式得,a,n,1.,等比数列的定义、通项公式、性质、前,n,项和公式是高考的热点内容，其中等比数列的基本量的计算能很好地考查考生对上述知识的应用以及对函数与方程、等价转化、分类讨论等思想方法的运用，是高考的一种重要考向,(3),通项公式法：若数列通项公式可写成,a,n,c,q,n,1,(,c,，,q,均为不为,0,的常数，,n,N,*,),，则,a,n,是等比数列,(4),前,n,项和公式法：若数列,a,n,的前,n,项和,S,n,k,q,n,k,(,k,为常数且,k,0,，,q,0,1),，则,a,n,是等比数列,4,等比数列的单调性,当,a,1,0,，,q,1,或,a,1,0,0,q,1,时为递增数列；当,a,1,0,，,q,1,或,a,1,0,0,q,1,时为递减数列；当,q,0,时为摆动数列；当,q,1,时为常数列,1,(2010,辽宁高考,),设,S,n,为等比数列,a,n,的前,n,项和，已知,3,S,3,a,4,2,3,S,2,a,3,2,，则公比,q,(,),A,3 B,4,C,5 D,6,答案：,B,答案：,A,3,等比数列,a,n,中，,|,a,1,|,1,，,a,5,8,a,2,，,a,5,a,2,，则,a,n,(,),A,(,2),n,1,B,(,2),n,1,C,(,2),n,D,(,2),n,答案：,A,答案：,5,设,a,n,是正项等比数列，令,S,n,lg,a,1,lg,a,2,lg,a,n,n,N,*,.,如果存在互异正整数,m,、,n,，使得,S,n,S,m,.,则,S,m,n,_.,答案：,0,6,若数列,a,n,满足,a,1,1,，,a,n,1,pS,n,r,(,n,N,*,),，,p,，,r,R,，,S,n,为数列,a,n,的前,n,项和,(1),当,p,2,，,r,0,时，求,a,2,，,a,3,，,a,4,的值；,(2),是否存在实数,p,，,r,，使得数列,a,n,为等比数列？若,存在，求出,p,，,r,满足的条件；若不存在，说明理由,解：,(1),因为,a,1,1,，,a,n,1,pS,n,r,，,所以当,p,2,，,r,0,时，,a,n,1,2,S,n,，,所以,a,2,2,a,1,2,，,a,3,2,S,2,2(,a,1,a,2,),2(1,2),6,，,a,4,2,S,3,2(,a,1,a,2,a,3,),2(1,2,6),18.,(2),因为,a,n,1,pS,n,r,，所以,a,n,pS,n,1,r,(,n,2),，,所以,a,n,1,a,n,(,pS,n,r,),(,pS,n,1,r,),pa,n,，,即,a,n,1,(,p,1),a,n,，其中,n,2,，,所以若数列,a,n,为等比数列，则公比,q,p,10,，,所以,p,1,，,又,a,2,p,r,a,1,q,a,1,(,p,1),p,1,，故,r,1.,所以当,p,1,，,r,1,时，数列,a,n,为等比数列,点击此图片进入课下冲关作业,