等差数列前n项和性质及应用

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,前,100,个自然数的和：,1+2+3+100=,；,前,n,个奇数的和：,1+3+5+(2,n,-1)=,；,前,n,个偶数的和：,2+4+6+2,n,=,.,思考题：,如何求下列和？,n,2,n,(,n,+1),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,【,说明,】,推导等差数列的前,n,项和公式的方法叫,；,等差数列的前,n,项和公式类同于,；,a,n,为等差数列,，这是一个关于,的没有,的“,”,倒序相加法,梯形的面积公式,S,n,=,an,2,+bn,n,常数项,二次函数,（注意,a,还,可以是,0,）,例,1,已知数列,a,n,中,S,n,=2,n,2,+3,n,，,求证：,a,n,是等差数列,.,例,1,、若等差数列,a,n,前,4,项和是,2,，前,9,项和是,6,，求其前,n,项和的公式。,，,解之得：,解：设首项为,a,1,，,公差为,d,，,则有：,设,S,n,=an,2,+,bn,，,依题意得：,S,4,=2,S,9,=,6,即,解之得：,另解：,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,前,n,项和为,S,n,，,S,m,=,S,l,问,:n,为何值时，,S,n,最大？,例,1,的,变式题一,：等差数列,a,n,中，首项,a,1,，,S,3,=S,11,，,问：这个数列的前几项的和最大,？,例,2,：已知数列,a,n,是等差数列，且,a,1,=21,，,公差,d=,2,，,求这个数列的前,n,项和,S,n,的最大值。,的前,n,项和为,当,n,为何值时，,最大，,数列,的通项公式,已知,求：,例,3,设等差数列,求等差数列前,n,项的最大,(,小,),的方法,方法,1:,由 利用二次函数的对称轴求得最值及取得最值时的,n,的值,.,方法,2:,利用,a,n,的符号当,a,1,0,d,0,时,数列前面有若干项为正,此时所有正项的和为,S,n,的最大值,其,n,的值由,a,n,0,且,a,n+1,0,求得,.,当,a,1,0,时,数列前面有若干项为负,此时所有负项的和为,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,1:S,n,S,2n,S,n,S,3n,S,2n,也成等差数列,公差为,在等差数列,a,n,中,其前,n,项的和为,S,n,则有,性质,2:,若,S,m,=,p,S,p,=,m(mp,),则,S,m+p,=,性质,3:,若,S,m,=S,p,(,mp,),则,S,p+m,=,n,2,d,0,-(,m+p,),性质,4:,为等差数列,.,两等差数列前,n,项和与通项的关系,性质,6:,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,例,1.,设等差数列,a,n,的前,n,项和为,S,n,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,其中项数为偶数的项的和与项数为奇数的项的和之比为,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,S,13,0,。,(1),求公差,d,的取值范围,;,(2),指出,S,1,S,2,S,12,中哪个值最大，,作业：,1:,等差数列,a,n,的前项和,S,n,满足,S,5,=95,S,8,=200,求,S,n,。,2:,若数列,a,n,的前项和,S,n,满足,S,n,=an,2,+bn,试判断,a,n,是否是等差数列,。,设,S,n,=an,2,+bn,则有：。,解之得：，,S,n,=3n,2,+n,。,1,、,2,、是。,简单提示：利用公式：,3,、,（,1,），（,2,）,S,6,最大。,