等差数列前n项和的性质.ppt

单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,单击此处编辑母版标题样式,等差数列的前,n,项和,的性质及应用,等差数列的前,n,项和公式,:,形式,1:,形式,2:,复习回顾,1.,将等差数列前,n,项和公式,看作是一个关于,n,的函数，这个函数,有什么特点？,当,d,0,时,S,n,是常数项为零的二次函数,则,S,n,=An,2,+Bn,令,等差数列的前,n,项的最值问题,例,1.,已知等差数列,a,n,中,a,1,=13,且,S,3,=S,11,求,n,取何值时,S,n,取最大值,.,解法,1,由,S,3,=S,11,得,d,=,2,当,n=7,时,S,n,取最大值,49等差数列的前,n,项的最值问题,例,1.,已知等差数列,a,n,中,a,1,=13,且,S,3,=S,11,求,n,取何值时,S,n,取最大值,.,解法,2,由,S,3,=S,11,得,d,=,20,当,n=7,时,S,n,取最大值,49.,则,S,n,的图象如图所示,又,S,3,=S,11,所以图象的对称轴为,7,n,11,3,S,n,等差数列的前,n,项的最值问题,例,1.,已知等差数列,a,n,中,a,1,=13,且,S,3,=S,11,求,n,取何值时,S,n,取最大值,.,解法,3,由,S,3,=S,11,得,d,=,2,当,n=7,时,S,n,取最大值,49.,a,n,=13+n-1)(-2)=,2n+15,由,得,a,7,+,a,8,=0,等差数列的前,n,项的最值问题,例,1.,已知等差数列,a,n,中,a,1,=13,且,S,3,=S,11,求,n,取何值时,S,n,取最大值,.,解法,4,由,S,3,=S,11,得,当,n=7,时,S,n,取最大值,49.,a,4,+,a,5,+,a,6,+,a,11,=0,而,a,4,+,a,11,=,a,5,+,a,10,=,a,6,+,a,9,=,a,7,+,a,8,又,d,=,20,a,7,0,a,8,0,d,0,时,此时,S,n,有,最大值,其,n,的值由,a,n,0,且,a,n+1,0,求得,.,当,a,1,0时,此时,Sn,的,最小值,其,n,的值由,a,n,0,且,a,n+1,0,求得,.,练习,:,已知数列,a,n,的通项为,a,n,=26-2n,要使此数列的前,n,项和最大,则,n,的值为,(),A.12 B.13 C.12,或,13 D.14,C,2.,等差数列,a,n,前,n,项和的性质,性质,2,:S,n,S,2n,S,n,S,3n,S,2n,也是等差数列,公差为,在等差数列,a,n,中,其前,n,项的和为,S,n,则有,n,2,d,性质,1,:,为等差数列,.,性质,3,:,若数列,a,n,与,b,n,都是等差数列,且前,n,项的和分别为,S,n,和,T,n,则,（,2),若项数为奇数,2n,1,则,S,2n-1,=(2n,1),a,n,(a,n,为中间项,),此时有,:S,奇,S,偶,=,a,n,性质,5,:,若,S,m,=p,S,p,=m(mp),则,S,m+p,=,(m+p),性质,6,:,若,S,m,=S,p,(mp),则,S,p+m,=,0,性质,4,:(1),若项数为偶数,2n,则,S,2n,=n(,a,1,+,a,2n,)=n(,a,n,+,a,n+1,)(,a,n,a,n+1,为中间项,),此时有,:S,偶,S,奇,=,nd,例,1.,设等差数列,a,n,的前,n,项和为,Sn,若,S,3,=9,S,6,=36,则,a,7,+,a,8,+,a,9,=()A.63 B.45 C.36 D.27,例,2.,在等差数列,a,n,中,已知公差,d=1/2,且,a,1,+,a,3,+,a,5,+,a,99,=60,a,2,+,a,4,+,a,6,+,a,100,=(),A.85 B.145 C.110 D.90,B,A,3.,等差数列,a,n,前,n,项和的性质的应用,例,3.,一个等差数列的前,10,项的和为,100,前,100,项的和为,10,则它的前,110,项的和为,.,110,例,4.,两等差数列,an,、,bn,的前,n,项和分别是,Sn,和,Tn,且,求 和,.,等差数列,a,n,前,n,项和的性质的应用,例,5.,一个等差数列的前,12,项的和为,354,其中项数为偶数项的和与项数为奇数项的和之比为,32:27,则公差为,.,例,6.(09,宁夏,),等差数列,a,n,的前,n,项的和为,S,n,已知,a,m-1,+,a,m+1,-,a,m,2,=0,S,2m-1,=38,则,m=,.,例,7.,设数列,a,n,的通项公式为,a,n,=2n-7,则,|,a,1,|+|,a,2,|+|,a,3,|+|,a,15,|=,.,5,10,153,等差数列,a,n,前,n,项和的性质的应用,例,8.,设等差数列的前,n,项和为,S,n,已知,a,3,=12,S,12,0,S,13,0,13,a,1,+136,d,0,等差数列,a,n前,n,项和的性质,(2),Sn,图象的对称轴为,由,(1),知,由上得,即,由于,n,为正整数,所以当,n=6,时,S,n,有最大值,.,S,n,有最大值,.,练习,1,已知等差数列,25,21,19,的前,n,项和为,S,n,求使得,S,n,最大的序号,n,的值,.,练习,2:,求集合,的元素个数，并求这些元素的和,.,练习,3,：已知在等差数列,a,n,中,a,10,=23,a,25,=-22,S,n,为其前,n,项和,.,（,1,）问该数列从第几项开始为负？,（,2,）求,S,10,（,3,）求使,S,n,0,的最小的正整数,n,.,(4),求,|,a,1,|+|,a,2,|+|,a3,|+,+|,a,n,|,的值,课堂小结,1.,根据数列前,n,项和，求通项公式,.,2,、结合二次函数图象和性质求,=An,+Bn,的最值,.,2.,等差数列,a,n,前,n,项和的性质,性质,2,:S,n,S,2n,S,n,S,3n,S,2n,也是等差数列,公差为,在等差数列,a,n,中,其前,n,项的和为,S,n,则有,n,2,d,性质,1,:,为等差数列,.,性质,3,:,若数列,a,n,与,b,n,都是等差数列,且前,n,项的和分别为,S,n,和,T,n,则,（,2),若项数为奇数,2n,1,则,S,2n-1,=(2n,1),a,n,(a,n,为中间项,),此时有,:S,奇,S,偶,=,a,n,性质,5,:,若,S,m,=p,S,p,=m(mp),则,S,m+p,=,(m+p),性质,6,:,若,S,m,=S,p,(mp),则,S,p+m,=,0,性质,4,:(1),若项数为偶数,2n,则,S,2n,=n(,a,1,+,a,2n,)=n(,a,n,+,a,n+1,)(,a,n,a,n+1,为中间项,),此时有,:S,偶,S,奇,=,nd,