(完整版)数列知识点总结及题型归纳

让学习成为一种习惯！一、数列的概念a1n2nna；na，a，aaa。123nna}nnn51111，，，，2345…a=n（nnNn1a=（nNnn①aana=fnnnnnk1(kZ)；②1)=nnkn项2536475869f(n)Nn从1ffff(n)fnan2n1an……S(n1)a1aS(n≥2)nSaSnnnnnn1a}n项和s2n3，求数列a}2nnn12222，，，；，，，。n2a3nna，，a，a，a；a2379）为.2让学习成为一种习惯！二、等差数列2个数列就叫等差数列，这个常数叫做等差数列的公差，公差通常用字母d表示。用递推公式表示为aad(n2)或aad(n1)。nn1an1n2n1aa，nnn1aa(nd；n10d0d0APdaa16，a，则aa）79412na}a1d3a2005nn1na2nb2n1a为nb为nnnaba，A，bAa与bA2aba，A，bA2aaa（2aaa）2n1nn2nnmnmaaaa15aaa80aaa（）111231231213nA．12010590752.设数列a}）na2nanaa(mn)；mam，nNaa(nm)dd，，nnmnmnm，n，p，qN且mnpqaaaa；anmnpqn(aa)n(n1)1dnSnandn（a）n2。12222n11)(SAnBn(,B为常数)a2nn(aa)n(aa)nSn(m1nm22naa12aa...aaan3451273让学习成为一种习惯！Sana，3a11S)7nn2672aaa=anSSn9249naaa10a）519n33）项项项项21，则aaaaanSSnn258SnSa则a5a95n53Snbbbbbn112bb；nnaa10S70d)n21132333s12aasnna63nnSaSanSSTn｝nnn7nnnT。nanSa30，a50nn1020aSnnna}S48,S168,求a和da10,S5,求a和S；(3)n81216588aa40,求S315174让学习成为一种习惯！偶奇；②SSa2nSSnd；奇偶nan1Sn2n1奇偶SSaa。奇n1n中S偶S,SS,SSn2nn3n2nam2m3m）nn2n3n。an14，SS30，则SanS=Snn4791SSSan＝＝363nnSS6313181910aad常数）（nNan1nn2aaa（nN)an1nn2naknb(k,b为常数)annnSAnBn(,B为常数)a2nna}aa2a}为（）nnn1na}a2n5a}为（）nnn5让学习成为一种习惯！a}的前ns2n4，则数列a}）2nnna}的前ns2n2，则数列a}）nnna}a2a0，则数列a}）an2n1nnn，且a2aa0（nN）aaan14n2n1nanSanSa）nnnn0，d0a01，d0aSS1nnSS，SSnanbn2nnnnNana0a0或aSnnNn。na0a0n1nnn1aa，SSn19anSnna12，S，S03dS，S，，S126让学习成为一种习惯！aSnSSSSS*nn566780）a0C.SSS与SS79567nn98a（nNan99nn5.已知a}a311d8。n（1）数列a}na}n前nna}a010S0a}n前d10nna}a25SSS，1179nnS(n1)a1SS(n2)nnn11.数列a}nSn15a}2nnn列a}nn4n，则anS2nn3.设数列a}nS2a}nnn1aa前n和S(na1n1n2nan7让学习成为一种习惯！ana}Sn2a）8n项和nn(qaq(q0)。a：qn1n递推关系：aaqn1n通项公式：aaqn1n1推广：aaqnmnm1．aaq2a1nn2．aaq2a_____.3719naaaq）n212，a54a=aan258a}a3aaa（）n1345ABCD,b,ca与c，注：acacab为bb22323和)()1(B)1(C)1(D)22a,a,aaa0a且n项1136nn和S）nn7nnnnn2222nn4433248让学习成为一种习惯！若mnpqaaaa(其中m,n,p,qN)mnpqanm(nN)q，aaa2nanmnmnmana既是等差数列又是等比数列ann2x5x10aa()2a和a1an1047()52C)1(D)1B()2222100a=aa，aa5n19a，aa，aaaan1634nn1anTlgalgalga,求Tn12nna}aaaa18,则logalogaloga（）n56473132310log53a}a0,n1,2,naa52(n3)2n1nn2n5logalogaloga212322n1（）n(2n(n2(n22n2.前nna(qq)aaq(1Saqnn11n1q1qa}a5q2nSn1na}a51q12nn项n和Sn9让学习成为一种习惯！a}a6aa30nSna和Sn(nN)f(n)）213nnf(n)2222247103n102222(81)(81)(81)(81)771737n4nnnanSSSS；nn369a}nSSqnnn.nan，S，Snkk3kS2kSkN*S2kSS3kaaaaaaaa123kk12k2k1SS3kSSSk2kk3k2kS9SSS63anSnn=673832n2n3n）aS10，S30，则Snm2m3maq（常数）n1aanna2aa(aan1nn2nnakq(k,q为常数）annnSkq)（k,q为常数）annnnSkkq（k,q为常数）annna}a2n，则数列a}为（）nnn2.已知数列a}aaa(a，则数列a}为（）2n1nn2nnn10让学习成为一种习惯！a}s22n1a}）的前nnnnS(n1)a1SS(n2)nnn11anSaaSaaa3nn1n1234nan5,SSn)aa前SnnN*1nn1nn列a1na}a7,aa26，求a；nn357a}a2,aana}n1nn1na，且a2aa0（nNaaa14n2n1nnn11a}a12aaannn1n11让学习成为一种习惯！11a0设数列a}11，求a}且1a1annn1n2a已知数列a}an1,a1，求数列a}na2n1nn等比数列a}2aa1a9aa，求数列a}，212326nn已知数列a}a2,a3a(n，求数列a}n1nn1n已知数列a}a，且aaaa2（nNa12n2nn1nn已知数列a}a，a52(a5)（nN且n1an11nnnna}a，a5223(a522)（nN且n1an11nnnn12让学习成为一种习惯！1aa,a4an1).a2n1nn1naaf(n)n1naaf21aaf若aaf(n)(n32n1naaf(n)n1nnaf(n)an11k111已知数列a}a,aa，求数列a}2n41n1n1n2n已知数列a}aa2n，a1，求数列a}nn1n1n已知数列a}aa23，a3，求数列a}nnn1n1n13让学习成为一种习惯！设数列a}a2aa32，，求数列a}2n111nnnn（af(n)an1naf(n)af，f，，f(n)aa若n12a13n1aaan2nanaf(k)n1a11k1a}a2(n1)5a，a3，求数列a}nnn1n1n2naa，an1aa。n13n1nnn1n2a3a，a(a。n1n1nnaqaf(n)n1nf(n)14让学习成为一种习惯！()afnn1([()]afnafnn112n2，12()fnan1ana}a1,a2an2)an1nn1naa1,a2a3(n1)a1n1nnn理aa1,a2anN).a*1n1nnn15让学习成为一种习惯！已知数列a}a2a35，a6annn1n1n1ax52(ax5)nnn1na}aa52，a1，求数列a}nnn1n1n1ax2y3(ax2y)nnn1n511,n1aaaaa()16n13n2nn已知数列a}a2a3n4n，a1，求数列a}2nn1n1n22ax(n1)y(n1)z2(axnynz)n1n已知数列a}a2a43，a1n1ann1n1n16让学习成为一种习惯！apaqaqn2n1nasat(asa)n2n1n1nstptstq已知数列a}a5a6a,a1,a2，求数列a}nn2n1n12n（SnS,n1a1a或SnSS,n2nnnn11anSaaSaaa3nn1n1234nan17让学习成为一种习惯！15,S)aa前nSSnnNa*1nn1nnn1前n和S(na1aan1n2nanan1已知数列a}nS(aa2)a,a,aS6nnnnn249列a}nn与1n151例a}aaCC,b，求数列b}2a2n1n1annnn18让学习成为一种习惯！1n1a1,a)aa}2n1n1nnnannbb{}nn（2aa}a,a1，求数列a}na2nn11nn1nn与I第a}na，aa2aa(na(n2)，求a}1n123n1nn3，a3a…3aaN*aaa2n1123nnn19让学习成为一种习惯！（已知数列a}aa3(n1)2n，5，求数列a}an1n1nnaa3(n1)2nn1naa3n2n1[a232a3(2nn2n2nn12nn(n2)(n1)n2nn1[a]3(n2)2n33(nn2(2)(1)n32nna3(n2)(n1)n2(n3)(n2)(n1)n3an131(n2)(nn2nnn(3)(2)(1)n(n1)an1n221n(n又a。nn5a}a531221nn（a}a23aa7，求数列a}，n5nn1n1na23a，a75a，a0。nn1n1nn1lga5lganlg3lg2n1n20让学习成为一种习惯！1a}a4a124aa1a}16n1nn1nn1b124aab1)224nnnnna(qqn)1qn(aa)(nSnandSa11n1n(q22n1a1S1qa}aa3n项和S}12nnaaaanS）n135na}aa3n项和S}12nnfn）()f(n)22222(nN)47103n1021让学习成为一种习惯！2272727n13n4nn7ab求ababab.,,nn1122nnS12x3xnxn12n123nSaaa3ann23.设a}b}ab1ab21ab13，，nn113553aa}b}S求，n．nbnnnn111111((2nn22n12n11n(nnn1)111(11111)[]n(n2)2nn2n(nn2)2n(n(nn2)n11n!(n!!CCCi1ii(n!(nn1nn11annaann11例：1.数列a}annSS）n(nnn522让学习成为一种习惯！156161a}ann(nnn1a}annn1nnn1＝111a}aTT．2aaaaaannnn1324nn21111,()。234naa1aabaanNblg()nnnnnS。nn23让学习成