等差數(shù)列及其前n項(xiàng)和全面總結(jié)課件

§6.2等差數(shù)列及其前n項(xiàng)和基礎(chǔ)知識(shí)自主學(xué)習(xí)要點(diǎn)梳理1.等差數(shù)列的定義如果一個(gè)數(shù)列從第二項(xiàng)起每一項(xiàng)與它相鄰前面一項(xiàng)的差是同一個(gè)常數(shù),那么這個(gè)數(shù)列就叫做等差數(shù)列,這個(gè)常數(shù)叫做等差數(shù)列的公差,通常用字母d表示2等差數(shù)列的通項(xiàng)公式如果等差數(shù)列{an的首項(xiàng)為a1公差為d,那么它的通項(xiàng)公式是a=a1+(n-1)d.3.等差中項(xiàng)A-4+6如果2,那么A叫做a與b的等差中項(xiàng)4等差數(shù)列的常用性質(zhì)(1)通項(xiàng)公式的推廣:an=an+(mn-m)d,(n,m∈N*)(2)若{an}為等差數(shù)列,且計(jì)+=m+n,(死,l,mn∈N*),則astaNata(3)若{an}是等差數(shù)列,公差為d,則{a2}也是等差數(shù)列,公差為2d(4)若an,是等差數(shù)列,則{pan+qb是等差數(shù)列(5)若{an}是等差數(shù)列,則ak,a+m,a+2m,…(,m∈N*)是公差為md的等差數(shù)列5.等差數(shù)列的前n項(xiàng)和公式n(a+an)設(shè)等差數(shù)列{an的公差為d,其前n項(xiàng)和Snn(n或Snm+26.等差數(shù)列的前n項(xiàng)和公式與函數(shù)的關(guān)系S=2n+(a-on數(shù)列{an}是等差數(shù)列的充要條件是其前n項(xiàng)和公式S∫(n)是m的二次函數(shù)或一次函數(shù)且不含常數(shù)項(xiàng),即Sn=An2+Bm,(A2+B2≠0)7在等差數(shù)列{an}中,a1>0,d<0,則S存在最大值;若a1<0,d>0,則S存在最小值8.等差數(shù)列與等差數(shù)列各項(xiàng)的和有關(guān)的性質(zhì)1)若{a是等差數(shù)列,則S也成等差數(shù)列,其首項(xiàng)與{an首項(xiàng)相同,公差是{an}公差的_2(2)Sn,S2n,S3n分別為{an}的前m項(xiàng),前2m項(xiàng),前3m項(xiàng)的和,Sn,S2nSn,S3nS2成等差數(shù)列(3)關(guān)于等差數(shù)列奇數(shù)項(xiàng)與偶數(shù)項(xiàng)的性質(zhì)①若項(xiàng)數(shù)為2n,則S偶S奇=nd②若項(xiàng)數(shù)為2n-1,則S偶(n-1)an,S奇幾anS奇S=1(4)兩個(gè)等差數(shù)列{an}、的前n項(xiàng)和Sn、Tn之間的關(guān)系為:基礎(chǔ)自測(cè)1.(2009·遼寧){an}為等差數(shù)列,且a72a4=-1,a3=0,則公差d(B)A.-2BcD.22解析根據(jù)題意得a-2a=a1+6d2(a1+3d=1,a1=1.又∵a2=a1+24=0,∴d2已知數(shù)列{an}中,a1=1,=+,則a10等于(B)1+1aABcD.以上都不對(duì)解析由a=,1=1+1得1為等差數(shù)列11l12+in=-n+03.(2009·福建)等差數(shù)列{an}的前n項(xiàng)和為S,且S32=6,a3=4,則公差d等于(C)A.1BC.2D.3解析設(shè){an}首項(xiàng)為a1,公差為d則s3=3a2+3d=3a1+3d=6,a32=a1+2d=4,∴a1=0,d24.已知等差數(shù)列{an}的前13項(xiàng)之和為39,則a+an+a等于(B)A.6B.9C.12D.18解析由S3=13(a1+3)13a7=39得a7=3,a6+a7+a8=3a5設(shè)S是等差數(shù)列{a}的前n項(xiàng)和,若等于(A)