高中数学专题复习课件：等比数列 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,*,等比数列,一、概念与公式,1.,定义,2.,通项公式,3.,前,n,项和公式,二、等比数列的性质,1.,首尾项性质,:,有穷等比数列中,与首末两项距离相等的两项积相等,即,:,特别地,若项数为奇数,还等于中间项的平方,即,:,a,1,a,n,=,a,2,a,n,-,1,=,a,3,a,n,-,2,=,.,若数列,a,n,满足,:,=,q,(,常数,),则称,a,n,为等比数列,.,a,n,+1,a,n,a,n,=,a,1,q,n,-,1,=,a,m,q,n,-,m,.,na,1,(,q,=1);,S,n,=a,1,-,a,n,q,1,-,q,=(,q,1).,a,1,(1,-,q,n,),1,-,q,a,1,a,n,=,a,2,a,n,-,1,=,a,3,a,n,-,2,=,=,a,中,2,.,特别地,若,m,+,n,=2,p,则,a,m,a,n,=,a,p,2,.,2.,若,p,+,q,=,r,+,s,(,p,、,q,、,r,、,s,N,*,),则,a,p,a,q,=,a,r,a,s,.,3.,等比中项,如果在两个数,a,、,b,中间插入一个数,G,使,a,、,G,、,b,成等比数列,则,G,叫做,a,与,b,的等比中项,.,5.,顺次,n,项和性质,4.,若数列,a,n,是等比数列,m,pn,成等差数列,则,a,m,a,p,a,n,成等比数列,.,6.,若数列,a,n,b,n,是等比数列,则数列,a,n,b,n,也是等比数列,.,a,n,b,n,G,=,ab,.,若,a,n,是公比为,q,的等比数列,则,a,k,a,k,a,k,也成等比数列,且公比为,q,n,.,k,=2,n,+1,3,n,k,=1,n,k,=,n,+1,2,n,7.,单调性,8.,若数列,a,n,是等差数列,则,b,a,n,是等比数列,;,若数列,a,n,是正项等比数列,则,log,b,a,n,是等差数列,.,三、判断、证明方法,1.,定义法,;,2.,通项公式法,;,3.,等比中项法,.,a,1,0,q,1,a,1,0,0,q,0,0,q,1,a,1,1,a,n,是递减数列,;,q,=1,a,n,是常数列,;,q,0,),的等比数列,.(1),求使,a,n,a,n,+1,+,a,n,+1,a,n,+2,a,n,+2,a,n,+3,(,n,N,*,),成立的,q,的取值范围,;(2),若,b,n,=,a,2,n,-,1,+,a,2,n,(,n,N,*,),求,b,n,的通项公式,.,(1),0,q,0.,后三数成等比数列,其最后一个数是,25,解得,:,a,=16,d,=4.,故所求四数分别为,12,16,20,25.,a,-,d,+,a,+,a,+,d,=48,且,(,a,+,d,),2,=25,a,.,a,-,d,=12,a,+,d,=20.,课后练习题,2.,在等比数列,a,n,中,a,1,+,a,6,=33,a,3,a,4,=32,a,n,+1,a,n,.(1),求,a,n,;(2),若,T,n,=lg,a,1,+lg,a,2,+,+lg,a,n,求,T,n,.,解,:,(1),a,n,是等比数列,a,1,a,6,=,a,3,a,4,=32.,又,a,1,+,a,6,=33,a,1,a,6,是方程,x,2,-,33,x,+32=0,的两实根,.,a,n,+1,a,n,a,6,0,b,n,0,由,式得,a,n,+1,=,b,n,b,n,+1,.,(1),证,:,依题意有,:2,b,n,=,a,n,+,a,n,+1,a,n,+1,=,b,n,b,n,+1,.,2,2,2,2,从而,当,n,2,时,a,n,=,b,n,-,1,b,n,代入,得,2,b,n,=,b,n,-,1,b,n,+,b,n,b,n,+1,.,2,2,b,n,=,b,n,-,1,+,b,n,+1,(,n,2).,b,n,是等差数列,.,(2),解,:,由,a,1,=1,b,1,=,2,及,两式易得,a,2,=3,b,2,=2.,3,2,从而,b,n,=,b,1,+(,n,-,1),d,=,(,n,+1).,2,2,故,a,n,+1,=(,n,+1)(,n,+2).,1,2,a,n,=,n,(,n,+1)(,n,2).,1,2,而,a,1,=1,亦适合上式,a,n,=,n,(,n,+1)(,n,N,*,).,1,2,S,n,=,=2(1,-,+,-,+,+,-,),n,1,1,2,1,2,1,3,1,n,+1,2,n,n,+1,=,.,8.,设数列,a,n,的前,n,项和为,S,n,(,其中,n,N,*,),若,S,n,=(,c,+1),-,ca,n,其中,c,为不等于,-,1,和,0,的常数,.(1),求证,a,n,是等比数列,;(2),设数列,a,n,的公比,q,=,f,(,c,),数列,b,n,满足,:,b,1,=,b,n,=,f,(,b,n,-,1,),(,其中,n,N,*,且,n,2,),.,求数列,b,n,的通项公式,.,1,3,(1),证,:,S,n,=(,c,+1),-,ca,n,(,n,N,*,),a,1,=(,c,+1),-,ca,1,.,(,c,+1),a,1,=,c,+1.,c,-,1,a,1,=1.,当,n,2,时,a,n,=,S,n,-,S,n,-,1,=,ca,n,-,1,-,ca,n,(,c,+1),a,n,=,ca,n,-,1,.,a,n,a,n,-,1,c,c,+1,=,这是一个与,n,无关的常数,.,c,-,1,且,c,0,(2),解,:,由,(1),知,q,=,f,(,c,)=,c,c,+1,b,n,=,f,(,b,n,-,1,)=,(,n,N,*,且,n,2).,b,n,-,1,b,n,-,1,+1,b,n,-,1,1,b,n,1,=+1.,b,n,-,1,1,b,n,1,即,-,=1.,是以,=3,为首项,1,为公差的等差数列,.,b,n,1,b,1,1,b,n,1,=3+(,n,-,1),1=,n,+2.,a,n,是以,1,为首项,为公比的等比数列,.,c,+1,c,b,n,=,.,n,+2,1,