单击此处编辑母版标题样式,单击此处编辑母版文本样式,二级,三级,四级,五级,*,最新中小学教学课件,*,4.2,数列解答题,高考命题规律,1,.,高考命题的完全考题,常与解三角形解答题交替在第,17,题呈现,.,2,.,解答题,12,分,中档难度,.,3,.,全国高考有,3,种命题角度,分布如下表,.,高考真题体验,对方向,新题演练提能,刷高分,命题角度,1,等差、等比数列的判定与,证明,1,.,(2016,全国,17),已知数列,a,n,的前,n,项和,S,n,=,1,+a,n,其中,0,.,(1),证明,a,n,是等比数列,并求其通项公式,;,高考真题体验,对方向,新题演练提能,刷高分,解,(1)(,等比数列的定义与通项公式,),高考真题体验,对方向,新题演练提能,刷高分,高考真题体验,对方向,新题演练提能,刷高分,高考真题体验,对方向,新题演练提能,刷高分,1,.,(2018,安徽江南十校,3,月联考,),已知,S,n,是数列,a,n,的前,n,项和,且满足,S,n,-,2,a,n,=n-,4,.,(1),证明,:,S,n,-n+,2,为等比数列,;,(2),求数列,S,n,的前,n,项和,T,n,.,(1),证明,原式转化为,S,n,-,2(,S,n,-S,n-,1,),=n-,4(,n,2),即,S,n,=,2,S,n-,1,-n+,4,所以,S,n,-n+,2,=,2,S,n-,1,-,(,n-,1),+,2,.,注意到,S,1,-,1,+,2,=,4,所以,S,n,-n+,2,为首项为,4,公比为,2,的等比数列,.,(2),解,由,(1),知,:,S,n,-n+,2,=,2,n+,1,所以,S,n,=,2,n+,1,+n-,2,于是,T,n,=,(2,2,+,2,3,+,+,2,n+,1,),+,(1,+,2,+,+n,),-,2,n,高考真题体验,对方向,新题演练提能,刷高分,高考真题体验,对方向,新题演练提能,刷高分,3,.,(2018,福建福州期末,),已知数列,a,n,的前,n,项和为,S,n,且,S,n,=,2,a,n,-,1,.,(1),证明数列,a,n,是等比数列,;,(2),设,b,n,=,(2,n-,1),a,n,求数列,b,n,的前,n,项和,T,n,.,解,(1),当,n=,1,时,a,1,=S,1,=,2,a,1,-,1,所以,a,1,=,1,当,n,2,时,a,n,=S,n,-S,n-,1,=,(2,a,n,-,1),-,(2,a,n-,1,-,1),所以,a,n,=,2,a,n-,1,所以数列,a,n,是以,a,1,=,1,为首项,以,2,为公比的等比数列,.,(2),由,(1),知,a,n,=,2,n-,1,所以,b,n,=,(2,n-,1)2,n-,1,所以,T,n,=,1,+,3,2,+,5,2,2,+,+,(2,n-,3)2,n-,2,+,(2,n-,1)2,n-,1,2,T,n,=,1,2,+,3,2,2,+,+,(2,n-,3)2,n-,1,+,(2,n-,1)2,n,-,得,-T,n,=,1,+,2(2,1,+,2,2,+,+,2,n-,1,),-,(2,n-,1)2,n,=,1,+,2,-,(2,n-,1)2,n,=,(3,-,2,n,)2,n,-,3,所以,T,n,=,(2,n-,3)2,n,+,3,.,高考真题体验,对方向,新题演练提能,刷高分,4,.,(2018,广西柳州、南宁第二次联考,),设,a,1,=,2,a,2,=,4,数列,b,n,满足,:,b,n+,1,=,2,b,n,+,2,且,a,n+,1,-a,n,=b,n,.,(1),求证,:,数列,b,n,+,2,是等比数列,;,(2),求数列,a,n,的通项公式,.,高考真题体验,对方向,新题演练提能,刷高分,又,b,1,=a,2,-a,1,=,4,-,2,=,2,b,1,+,2,=,4,b,n,+,2,是以,4,为首项,以,2,为公比的等比数列,.,(2),由,(1),可得,b,n,+,2,=,4,2,n-,1,故,b,n,=,2,n+,1,-,2,.,a,n+,1,-a,n,=b,n,a,2,-a,1,=b,1,a,3,-a,2,=b,2,a,4,-a,3,=b,3,a,n,-a,n-,1,=b,n-,1,.,累加得,a,n,-a,1,=b,1,+b,2,+b,3,+,+b,n-,1,a,n,=,2,+,(2,2,-,2),+,(2,3,-,2),+,(2,4,-,2),+,+,(2,n,-,2),即,a,n,=,2,n+,1,-,2,n,(,n,2),.,而,a,1,=,2,=,2,1,+,1,-,2,1,a,n,=,2,n+,1,-,2,n,(,n,N,*,),.,高考真题体验,对方向,新题演练提能,刷高分,等差、等比数列的通项公式与前,n,项和公式的应用,1,.,(2018,全国,17),记,S,n,为等差数列,a,n,的前,n,项和,已知,a,1,=-,7,S,3,=-,15,.,(1),求,a,n,的通项公式,;,(2),求,S,n,并求,S,n,的最小值,.,解,(1),设,a,n,的公差为,d,由题意得,3,a,1,+,3,d=-,15,.,由,a,1,=-,7,得,d=,2,.,所以,a,n,的通项公式为,a,n,=,2,n-,9,.,(2),由,(1),得,S,n,=n,2,-,8,n=,(,n-,4),2,-,16,.,所以当,n=,4,时,S,n,取得最小值,最小值为,-,16,.,高考真题体验,对方向,新题演练提能,刷高分,2,.,(2018,全国,17),等比数列,a,n,中,a,1,=,1,a,5,=,4,a,3,.,(1),求,a,n,的通项公式,;,(2),记,S,n,为,a,n,的前,n,项和,若,S,m,=,63,求,m.,解,(1),设,a,n,的公比为,q,由题设得,a,n,=q,n-,1,.,由已知得,q,4,=,4,q,2,解得,q=,0(,舍去,),q=-,2,或,q=,2,.,故,a,n,=,(,-,2),n-,1,或,a,n,=,2,n-,1,.,(2),若,a,n,=,(,-,2),n-,1,则,.,由,S,m,=,63,得,(,-,2),m,=-,188,此方程没有正整数解,.,若,a,n,=,2,n-,1,则,S,n,=,2,n,-,1,.,由,S,m,=,63,得,2,m,=,64,解得,m=,6,.,综上,m=,6,.,高考真题体验,对方向,新题演练提能,刷高分,3,.,(2016,全国,17),S,n,为等差数列,a,n,的前,n,项和,且,a,1,=,1,S,7,=,28,.,记,b,n,=,lg,a,n,其中,x,表示不超过,x,的最大整数,如,0,.,9,=,0,lg99,=,1,.,(1),求,b,1,b,11,b,101,;,(2),求数列,b,n,的前,1 000,项和,.,解,(1),设,a,n,的公差为,d,据已知有,7,+,21,d=,28,解得,d=,1,.,所以,a,n,的通项公式为,a,n,=n.,b,1,=,lg,1,=,0,b,11,=,lg,11,=,1,b,101,=,lg,101,=,2,.,高考真题体验,对方向,新题演练提能,刷高分,依题意有,n,2,-,2,n,35,解得,n,7,.,故使,S,n,35,成立的,n,的最小值为,8,.,高考真题体验,对方向,新题演练提能,刷高分,2,.,(2018,广东东莞二模,),已知等比数列,a,n,与等差数列,b,n,a,1,=b,1,=,1,a,1,a,2,a,1,a,2,b,3,成等差数列,b,1,a,2,b,4,成等比数列,.,(1),求,a,n,b,n,的通项公式,;,(2),设,S,n,T,n,分别是数列,a,n,b,n,的前,n,项和,若,S,n,+T,n,100,求,n,的最小值,.,高考真题体验,对方向,新题演练提能,刷高分,解,(1),设数列,a,n,的公比为,q,数列,b,n,的公差为,d,则,高考真题体验,对方向,新题演练提能,刷高分,3,.,(2018,四川南充三诊,),已知,a,n,是等比数列,a,1,=,2,且,a,1,a,3,+,1,a,4,成等差数列,.,(1),求数列,a,n,的通项公式,;,(2),若,b,n,=,log,2,a,n,求数列,b,n,前,n,项的和,.,解,(1),设数列,a,n,的公比为,q,则,a,3,=a,1,q,2,=,2,q,2,a,4,=a,1,q,3,=,2,q,3,因为,a,1,a,3,+,1,a,4,成等差数列,所以,a,1,+a,4,=,2(,a,3,+,1),即,2,+,2,q,3,=,2(2,q,2,+,1),整理得,q,2,(,q-,2),=,0,因为,q,0,所以,q=,2,所以,a,n,=,2,2,n-,1,=,2,n,(,n,N,*,),.,(2),因为,b,n,=,log,2,a,n,=,log,2,2,n,=n,高考真题体验,对方向,新题演练提能,刷高分,高考真题体验,对方向,新题演练提能,刷高分,5,.,(2018,安徽马鞍山第二次质监,),已知数列,a,n,是等差数列,其前,n,项和为,S,n,a,2,=,37,S,4,=,152,.,(1),求数列,a,n,的通项公式,;,(2),求数列,|a,n,-,2,n,|,的前,n,项和,T,n,.,解,(1),设数列,a,n,的首项为,a,1,公差为,d,高考真题体验,对方向,新题演练提能,刷高分,高考真题体验,对方向,新题演练提能,刷高分,6,.,(2018,河南焦作模拟,),已知,a,n,为等差数列,且,a,2,=,3,a,n,前,4,项的和为,16,数列,b,n,满足,b,1,=,4,b,4,=,88,且数列,b,n,-a,n,为等比数列,.,(1),求数列,a,n,和,b,n,-a,n,的通项公式,;,(2),求数列,b,n,的前,n,项和,S,n,.,高考真题体验,对方向,新题演练提能,刷高分,解,(1),设,a,n,的公差为,d,因为,a,2,=,3,a,n,前,4,项的和为,16,高考真题体验,对方向,新题演练提能,刷高分,一般数列的通项公式与前,n,项和的求解,1,.,(2017,天津,18),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,的等比数列,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,的通项公式,;,(2),求数列,a,2,n,b,2,n-,1,的前,n,项和,(,n,N,*,),.,高考真题体验,对方向,新题演练提能,刷高分,解,(1),设等差数列,a,n,的公差为,d,等比数列,b,n,的公比为,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,的通项公式为,a,n,=,3,n-,2,数列,b,n,的通项公式为,b,n,=,2,n,.,高考真题体验,对方向,新题演练提能,刷高分,(2),设数列,a,2,n,b,2,n-,1,的前,n,项和为,T,n,由,a,2,n,=,6,n-,2,b,2,n-,1,=,2,4,n-,1,有,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