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1、高中数学：数列通项公式的求法论文新课标人教B版必修5论数列通向公式的求法 摘要:数列在理论上和实践中均有较高的价值，是培养学生观察能力、理解能力、逻辑思维能力的绝好载体，高考对数列知识的考察在八十年代末发展到了极致，以后逐渐冷落，但最近几年又逐渐升温，随着与大学知识的接轨，竞赛题的释放，很多省市的高考数学卷都把数列题作为压轴题，而数列通向公式的求法又成为一个热点。本文想总结一下在高中阶段，求数列的通项公式的常用方法和策略。 关键词:数列 通向公式 递推公式 求法 1( 观察法 观察法就是观察数列特征，找出各项共同的构成规律，横向看各项之间的关系结构，纵向看各项与项数n的内在联系，从而归纳出数列
2、的通向公式，然后利用数学归纳法加以证明即可。 例1 在数列中且成等差数列，成等ab,bab,ab,2,4aba,nnnnn,11，11nnn，1*比数列。求及，由此猜测的通向公式，并证明ab,nN,aaa,bbb,，nn234234你的结论。 2解:有题设条件得， 2baa,，abb,nnn，1，nnn11由此得， aaa,6,12,20bbb,9,16,252342342annbn,，,，1,1 猜测 ，nn用数学归纳法证明: (1) 当n=1时，有以上知结论成立; 2akkbk,，,，1,1nk,，1 (2) 假设n=k时，结论成立;即，那么当，kk2a2k，1abakkkkk,，,，,，
3、221112时， ,，2，bk，kkk，1k，1bknk,，1所以当时，结论也成立， 2annbn,，,，1,1由(1)( 2) ，可知对一切正整数都成立1 ，nn点评:采用数学归纳法证明多是理科教学内容，较为容易，好掌握。 ( 定义法 2直接利用等差数列或等比数列的定义求通项的方法叫定义法，这种方法适应于已知数列类型的题目( saaaa例2 等差数列是递增数列，前n项和为，且成等比数列，,1,3,9nn2asa,(求数列的通项公式. ,n55a 解:设数列公差为d(d0) ,naaa?成等比数列， 1,3,92？,aaa 319点评:利用定义法求数列通项时要注意不用错定义，设法求出首项与公差
4、(公比)后再写出通项。 3(公式法 saaa,若已知数列的前n项和与的关系，求数列的通项可用公式nnnn求解。 aSann,，,21,1a 例3 已知数列的前n项和满足(求数列的S,，,nnnnn通项公式。 点评:利用公式求解时，要注意对n分类讨论，但若能合写时一定要合并(2 4(由递推公式求数列通项法 对于递推公式确定的数列的求解，通常可以通过递推公式的变换，转化为等差数列或等比数列问题，有时也用到一些特殊的转化方法与特殊数列。3 4(1 类型1 递推公式为 ，其中的和比较aafn,，()fffn(1)(2).()，nn，1易求 ，通常解法是把原递推公式转化为，利用累加法(逐差相加aafn,
5、()nn，1法)求解。 11例4 已知数列中，求的通向公式 aaa,，aa,11nn，nn2n,2411111,解: 由已知得， aa,nn，1,2nnn4122121,，,令，代入个等式累积，即 nn,1,2,.,1n,1，111111，, aaaaaa,，,，,，,，,.1.，21321nn,nn23352321,，11, ？,aa1n1,n,221,43n,？,a n42n,aafn,4(2 类型2 递推公式为。 ，nn，1an，1解法: (1)把原递推公式转化为，利用累乘法求解。 ,fn，an2n,aaaa,例5 已知数列满足，求的通向公式。 a,11nn，nn，31nann，1 解:
6、由条件知，分别令n=1，2，3，(n-1)，代入上式得(n-1),1，ann个等式累乘之，即 afna,aa(2)由和确定的递推数列的通项可如下求得: ，,nn，1n1afna,1afna,2afa,1由已知递推式有，依，nn,1nn,1221afnfnffa,12.21次向前代入，得，简记为，n1，这就是叠(迭)代法的基本模式。 31n,aaan,3,(1)11nn，an32n，例6 已知，求。 3(1)13(2)132131nn,，,aa,？n13(1)23(2)232232nn,，,，解: 3437526nn,？331348531nnn, 。4 pan4(3类型3 递推公式为(p,q,s
7、,为常数)。 ,an，1，qasn解法: 利用两边取倒数求通向公式。 3a3n例7 已知数列的a首项，求a的通向公式 ,aa,nn11，n，521an,111123a121n解 ，？,11,1， ？,？,，a,，1naaa33，2133aaann，11，1,nnn,1211, 是以为首项，为公比的等比数列 ,a33n,n31212a, ， ,1nnn,1na，32333n3xx,另解: 设，解得xx,0,1 1221x，3aa,1nn方法1:,，整理得， a11a,1，1，1nn，21a21a，nn21a，13n 则 ,，2aaa,111，1nnn,11，,，131 即 ,aa,1nn，,11
8、,13，1,故数列是以为首项，为3公比的等比数列 ,a2n,1,n313n,1a,，, 则，即 13nna，322n,13aa,1aa3nnn，1n方法2: 由，可得， ,aa,1，1，1nn，,21aaa1121a，nnn，1n,a3n故数列是以为首项，3为公比的等比数列 ,2an,1,nn3a3n 则，即。 a,nn，322a,1nsat，sxt，n点评:形如( 为常数)的数列可用方程解stpq,x,a,n，1pxq，paq，nsat，sat，nn得两根，然后利用或，直接整理xx,axx,axx,n，111n，12212paq，paq，nn转化求解，也可将两式作比进行求解，此种方法称为“特
9、征根法”。5 4(4类型4 递推公式为型的。 f(S,a),0nn解法:(1)用“退一相减法”;(2)利用;(3)归纳，a,S,S,(n,2)nnn,1猜想，证明。方法(1)和方法(2)的实质是由混合型的转化为纯粹型的，也就是“减元思想”的应用。 a 例8 已知数列的前n项和与之间满足Sa,nnnSa,，1nna2lglglg1,，,Sa，求的通向公式 ，,nnn22Sa,，1,nn解 ？lglg1,Sa，nn,，2,2 ？，,，SaSa141，nnnn，2Sa,1 ，Sa,10， (1) ，nnnn，下面用三种方法解答: Sa,1方法一:下标退一，可得 (2) nn,11aaaaa,，,2(
10、1)-(2)得 nnnnn,111ana,Sa,1,22即，由，得 n，1112a,1n11a,a数列是以为首项，以为公比的等比数列 ,1n22nn,1111, a,n222,11方法二:由得， SS,，SSS,1a,S,S,(n,2)，nn,1nnn,1nnn,1221S,11n可得， SS,11,，nn,12S,12,1n11数列是以为首项，以为公比的等比数列。 S,1S,1,1n22nn,1nn11111,n,2所以，则，当时， SS,aSS,11,1,nnnnn22222,n11,又适合此式，所以 aS,a,11,n22,n1111,方法三:易得aaa,，故猜想 a,123,n2482
11、,下面用数学归纳法证明(证略) 4(5类型5 递推公式为(其中p，q均为常数，)。 解法:一般采用待定系数法将原递推公式转化为:，其中，再利用换元法转化为等比数列求解。6 例9 已知数列中，求。 atat，,，2,解: 令 ，nn，1t,3与已知aa,，23比较，得 nn，1？，,，aa323a，3a，,34，所以，数列是以为首项，2为公比的等比数，,nn，1n1列 nn,，11n，1aa，,，,3322所以 即7 a,23，n1naaa4(6类型6 递推公式为=p+q (p、q均为常数)(又称二阶递归) n，2n，1naaaaaaa,解法: 将原递推公式=p+q，转化为-,(-)并n，2n，
12、2n，1nn，1n，1n,，,p,aa,且由解出、,因此可以得到数列,-,是等比数列。特殊地,n，1n,q,对于 型的递推公式，我们可以的这样分析: a,pa，qa，p，q,1n，2n，1n? p，q,1? p,1,q，a,1,qa，qa,a,qa,an，2n，1nn，1n，1n，a,a,qa,an，2n，1n，1na,an，2n，1 ,qa,an，1n?是以为首项，公比为的等比数列 ,qa,aa,an，1n21552例10 已知数列中a=1，a= =-,求数列,的通项公aaaa12n，2n，1nn333式。 an5,，,2,3解: 令-,(-)由解得:,1、, aaaa,n，2n，1n，1n
13、23,3,22则由此可得-,(-)， a-a= aaaa21n，2n，1n，1n33n,12,?-a= ann,1,3,n,1n,2n2222,aaa?a,(a-)+(-)+?+(a-a)+a=+?+1=3-. 211n,2nnn,1n,1,1,n,3333,n2?a,3- n1n,3总之，求数列通向公式的方法并不满足以上所述，对于同一问题的求解也不仅是一种方法，只有在平时学习与探究过程中不断地体会与总结，将知识与方法学活，才可以做到游刃有余。 参考文献 1高慧明.数列通项的求法在2008年高考中的展示.J试题与研究,2008,20. 2龙志明.数列通项公式的九种求法.J求学,2005,11.
14、 3陈云烽.递推数列通项的求解.J中学数学教学参考,2007,6. 4刘有路.叠加叠乘在高考数列解题中的应用.J试题与研究,2005,14. 5 高慧明.数列通项的求法在2008年高考中的展示.J试题与研究,2008,20. 6吴怀芳.求数列通项的几种常见类型.J试题与研究,2005,26. 7聂文喜.高考递推数列题型分类解析.J试题与研究,2006,26.(英文版 ) Two regulations promulgated for implementation is in the party in power for a long time and the rule of law condi
15、tions, the implementation of comprehensive strictly strategic plan, implementation in accordance with the rules and discipline to manage the party, strengthen inner-party supervision of major initiatives. The two regulations supporting each other, the adhere to a positive advocate, focusing on moral
16、ity is of Party members and Party leading cadres can see, enough to get a high standard; around the party discipline, disciplinary ruler requirements, listed as negative list, focusing on vertical gauge, draw the party organizations and Party members do not touch the bottom line . Here, the main fro
17、m four square face two party rules of interpretation: the first part introduces two party Revised regulations the necessity and the revision process; the second part is the interpretation of the two fundamental principles of the revision of laws and regulations in the party; the third part introduce
18、s two party regulations modified the main changes and needs to grasp several key problems; the fourth part on how to grasp the implementation of the two regulations of the party. and revised the necessity and revised history of the CPC Central Committee the amendment to the Chinese Communist Party m
19、embers and leading cadres honest politics several guidelines and column 1 by 2015 to strengthen party laws and regulations focus. Two party regulations revision work lasted a Years, pooling the wisdom of the whole party, ideological consensus, draw historical experience, respect for the wisdom of ou
20、r predecessors, which reflects the unity of inheritance and innovation; follow the correct direction, grasp the limited goals, adhere to the partys leadership, to solve the masses of the people reflect a focus on the problem. The new revision of the and , reflects the partys 18 and the eighth sessio
21、n of the third, the spirit of the fourth plenary session, reflecting the experience of studying and implementing the General Secretary Xi Jinping series of important speech, reflects the partys eighteen years comprehensive strictly practice. (a) revised two regulations of the party need of in based
22、on revised, the promulgation and implementation of January 2010, to strengthen the construction of the contingent of leading cadres play an important role. But with the party to manage the party strictly administering the deepening, has not been able to fully meet the actual needs. Content is too co
23、mplicated, eight prohibition, 52 are not allowed to hard to remember, and also difficult to put into practice; the second is concisely positive advocated by the lack of prohibited provisions excessive, no autonomy requirements; the third is banned terms and discipline law, both with the party discip
24、line, disciplinary regulations repeat and Criminal law and other laws and regulations repeat; the fourth is to clean the theme is not prominent, not for the existing problems, and is narrow, only needle of county-level leading cadres above. is in 1997 based on revision, in December 2003 the promulga
25、tion and implementation, to strengthen the construction of the party play very important role. Along with the development of the situation, which many provisions have been unable to fully meet the comprehensive strictly administering the practice needs. One is Ji law, more than half of the provision
26、s and criminal law and other countries laws and regulations Repetition; two is the political discipline regulations is not prominent, not specific, for violation of the party constitution, damage the authority of Party Constitution of misconduct lack necessary and serious responsibility to pursue; t
27、hird is the main discipline for the leading cadres, does not cover all Party members. Based on the above situation, need to and the two is likely to be more relevant regulations first amendment. By revising, really put the authority of Party discipline, the seriousness in the party tree and call up
28、the majority of Party members and cadres of the party constitution of party compasses party consciousness. (II) two party regulations revision process the Central Committee of the Communist Party of China attaches great importance to two regulations revision . Xi Jinping, general books recorded in t
29、he Fifth Plenary Session of the eighth session of the Central Commission for Discipline Inspection, on the revised regulations made clear instructions. According to the central deployment, the Central Commission for Discipline Inspection from 2014 under six months begin study two regulations revisio
30、n. The Standing Committee of the Central Commission for Discipline Inspection 4 review revised. Comrade Wang Qishan 14 times held a special meeting to study two regulations revision, amendment clarifies the direction, major issues of principle, path and target, respectively held a forum will listen
31、to part of the province (area) secretary of the Party committee, Secretary of the Discipline Inspection Commission, part of the central ministries and state organs Department The first party committee is mainly responsible for people, views of experts and scholars and grassroots party organizations
32、and Party members. Approved by the Central Committee of the Communist Party of China, on 7 September 2015, the general office of the Central Committee of the Party issued a notice to solicit the provinces (autonomous regions, municipalities) Party, the central ministries and commissions, state minis
33、tries and commissions of the Party (party), the General Political Department of the military, every 3 people organization of Party of two regulations revision opinion. Central Commission for Discipline Inspection of extensive solicitation of opinions, careful study, attracting, formed a revised sent
34、 reviewers. In October 8 and October 12, Central Committee Political Bureau Standing Committee and the Political Bureau of the Central Committee After consideration of the two regulations revised draft. On October 18, the Central Committee of the Communist Party of China formally issued two regulati
35、ons. Can say, two laws amendment concentrated the wisdom of the whole party, embodies the party. Second, and revision of the basic principles of two party regulations revision work and implement the partys eighteen, ten eight plenary, the spirit of the Fourth Plenary Session of the Eleventh Central
36、Committee and General Secretary Xi Jinping important instructions on the revised and , highlighting the ruling party characteristics, serious discipline, the discipline quite in front of the law, based on the current, a long-term, advance as a whole, with Bu Xiuding independent and . Main principle
37、is: first, adhere to the party constitution to follow. The constitution about discipline and self-discipline required specific, awaken the party constitution of party compasses party consciousness, maintaining the authority of the constitution. General Secretary Xi Jinping pointed out that no rules,
38、 no side round. Party constitution is the fundamental law, the party must follow the general rules. In early 2015 held the eighth session of the Central Commission for Discipline Inspection Fifth Plenary Session of the 16th Central Committee, Xi Jinping again pointed out that constitution is the par
39、ty must follow the general rules, but also the general rules. the revision of the and is Method in adhere to the regulations established for the purpose of combining rule of virtue is to adhere to the party constitution as a fundamental to follow, the constitution authority set up, wake up the party
40、 constitution and party rules the sense of discipline, the party constitution about discipline and self-discipline specific requirements. 4 second is to adhere to in accordance with the regulations governing the party and the party. The Party of rule of virtue de, mainly refers to the partys ideals
41、and beliefs, excellent traditional style. The revised the closely linked to the self-discipline, insisting on the positive initiative, for all members, highlight the vital few, emphasized self-discipline, focusing on the morality, and the majority of Party members and the ideological and moral stand
42、ards. The revised Ji method separately, Ji, Ji Yan to Method, as a negative list, emphasizing the heteronomy, focusing on vertical gauge. Is this one high and one low, a positive reaction, the strict party discipline and practice results transformation for the integration of the whole party to obser
43、ve moral and discipline requirements, for the majority of Party members and cadres provides benchmarking and ruler. Third, insist on to. In view of the problems existing in the party at the present stage, the main problems of Party members and cadres in the aspect of self-discipline and abide by the
44、 discipline to make clearly defined, especially the partys eighteen years strict political discipline and political rules, organization and discipline and to implement the central eight provisions of the spirit against the four winds and other requirements into Disciplinary provisions. Not one pace
45、reachs the designated position, focusing on in line with reality, pragmatic and effective. After the revision of major changes, major changes in the and modified and needs to grasp several key problems (a) adhere to according to regulations governing the party and party with morals in combination, for at the present stage, the leadership of the party members and cadres and Party members in existing main problems of self-discipline, put forward principles, requirements and specifications, showing Communists noble moral pursuit, reflected at all times and i