數(shù)列求和經(jīng)典例題

1、v1.0可編輯可修改數(shù)列通項(xiàng)的方法利用.觀察法求數(shù)列的通項(xiàng).G(n1)利用公式法求數(shù)列的通項(xiàng)：an；an等差、等比數(shù)列an公式.SnSn1(n2)nn應(yīng)用迭加(迭乘、迭代)法求數(shù)列的通項(xiàng)：an1anf(n);an1anf(n).構(gòu)造等差、等比數(shù)列求通項(xiàng)：r23n 2n n N ，求an的通項(xiàng)公式。an1panq;an1panqn;an1panf(n);an2pan1qan.示例已知下列各數(shù)列an的前n項(xiàng)和Sn的公式為Sn題型一利用公式法求通項(xiàng)例數(shù)列an的前n項(xiàng)和記為Sba1=1,an+1=2Sn+1(n>1).(1)求an的通項(xiàng)公式；(2)等差數(shù)列bn的各項(xiàng)為正數(shù)，前n項(xiàng)和為不，且T3=

2、15,又a1+b1,a?+b2,a3+b3成等比數(shù)列，求Tn.練3數(shù)列an是公差大于零的等差數(shù)列，a2,a5是方程x212x270的兩根。數(shù)列bn的前n項(xiàng)和為T(mén)n,1.，且Tn1-bnnN,求數(shù)列an,bn的通項(xiàng)公式。23.已知數(shù)列an中，a1=1,an1-an=an1-an,則數(shù)列通項(xiàng)an=。例已知an的首項(xiàng)ai1,anian2n,（nN*）,求an的通項(xiàng)公式，并求a的值。題型二應(yīng)用迭加（迭乘、迭代）法求通項(xiàng)練1數(shù)列an中，a11,ann（an1an）,則數(shù)列an的通項(xiàng)an（）A2n1B.n2C.（口）n1D.nn2 an,求數(shù)列 an的通項(xiàng)公式.n練2已知Sn為數(shù)列an的前n項(xiàng)和，a11,

3、Sn15例數(shù)列an中，an 13an2(nN),且a108,貝Ua()B.8081C.27D.2627題型三構(gòu)造等比數(shù)列求通項(xiàng)1練1數(shù)列an中，a11,anan11n2,求通項(xiàng)公式ano2例已知數(shù)列an中，a11,an12an3n,求數(shù)列an的通項(xiàng)公式的通項(xiàng)練2設(shè)數(shù)列an的前n項(xiàng)和為Sn,已知a1a,an1Sn3n(nN),設(shè)bnSn3n,求數(shù)列公式.數(shù)列求和方法1.基本r數(shù)列白前n項(xiàng)和n(aian)21等差數(shù)列an的刖n項(xiàng)和：Snnai-n(n1)d2anbn等比數(shù)列an的前n項(xiàng)和Sn:當(dāng)q 1時(shí)，Sn nai；當(dāng)q 1時(shí)，Sna1(1qn)aanq;1 q 1 q2.數(shù)列求和的常用方法：公

4、式法；性質(zhì)法；拆項(xiàng)分組法；裂項(xiàng)相消法；錯(cuò)位相減法；倒序相加法題型一公式法、性質(zhì)法求和1.已知Sn為等比數(shù)列an的前n項(xiàng)和，公比q2,S997,則a3a6a9a992.等差數(shù)列an中，公差d1 a3 a5a99則 aa2 a3a100111例1求數(shù)列1 ,2,3 ,2 4 8，(n12n，的前n項(xiàng)和Sn.題型二拆項(xiàng)分組法求和練2在數(shù)列an中，已知a1=2,an+1=4an-3n+1,neN.S,求(1)求數(shù)列an的通項(xiàng)公式；(2)設(shè)數(shù)列an的前n項(xiàng)和為練.求數(shù)列(2n1)2的前n項(xiàng)和Sn.1n(n 1)一一11例.求和：1223題型三裂項(xiàng)相消法求和1例.求和：一.2114.3練4已知數(shù)列an滿足

5、a11,an1*2an1nN（1）求數(shù)列an的通項(xiàng)公式。（2）若數(shù)列an1 n,求數(shù)列bn的通項(xiàng)公b12b913b1nbn1bn滿足4142434n式。（3）若cn的前n項(xiàng)和Sn。2n,求數(shù)列cnanan1【示例】以a1為首項(xiàng)等比數(shù)列，q為公比,n項(xiàng)和sn的推導(dǎo)題型四錯(cuò)位相減法求和例.設(shè)數(shù)列 an 為 1 2,2 22,3 23,4 232nx 0求此數(shù)列前n項(xiàng)的和,nan = -,3例.設(shè)數(shù)列an滿足ad3a2+32a3+3n-1n（1）求數(shù)列an的通項(xiàng)公式；（2）設(shè)bn=-,求數(shù)列bn的前n項(xiàng)和Sn.anbn1*、練1已知數(shù)列an、bn滿足a11,a23,2(nN),bnan1an。bn(

6、1)求數(shù)列bn的通項(xiàng)公式；*、(2)數(shù)列Cn滿足cnbnlog2(an1)(nN),求SnCiC2.Cn。的值練4等比數(shù)列an中，已知對(duì)任意自然數(shù)n,aia?a3an2n1,求a；a2a2a22A. 2n 1B.；2n 1C.4n 1課后練習(xí)的首項(xiàng)1設(shè)正項(xiàng)等比數(shù)列(I)求的通項(xiàng)；（H）求的前n項(xiàng)和2數(shù)列的前項(xiàng)和記為求的通項(xiàng)公式；3在數(shù)列an與bn中,ai=1,b1=4,數(shù)列an的前n項(xiàng)和Sn滿足nS+i-(n+3)Sn=0,nN.求a2,的值；(2)求數(shù)歹1an通項(xiàng)公式;4設(shè)數(shù)列an的前n項(xiàng)和為對(duì)任意的正整數(shù)n,都有a0501成立，記必Un(nN1an求數(shù)列bn的通項(xiàng)公式5設(shè)數(shù)列an的前n項(xiàng)和

7、為且對(duì)任意正整數(shù)n,an+S=4096.(1)求數(shù)列an的通項(xiàng)公式：設(shè)數(shù)歹Ilog2an的前n項(xiàng)和為T(mén)n.對(duì)數(shù)列Tn,從第幾項(xiàng)起Tn<-5096設(shè)數(shù)列an的前n項(xiàng)和(1)求首項(xiàng)ai求證是等比數(shù)列(2)求數(shù)列an的通項(xiàng)公式7(17)設(shè)等差數(shù)列an的前n項(xiàng)和為Sn,公比是正數(shù)的等比數(shù)列bn的前n項(xiàng)和為T(mén)n,已知ai1,n3自b317工12,求an,bn的通項(xiàng)公式.8已知數(shù)列an的前n項(xiàng)和Sn2n22n,數(shù)歹【bn的前n項(xiàng)和Tn2b(I)求數(shù)列an與bn的通項(xiàng)公式；(H)設(shè)Cnan2bn,證明：當(dāng)且僅當(dāng)n3時(shí)，Cn1g.9等比數(shù)列an的前n項(xiàng)和為Sn,已知對(duì)任意的nN，點(diǎn)(n,Sn),均在函數(shù)ybxr(b0且b1,b,r均為常數(shù))的圖