全国高中数学联赛之数列问题研究

发布时间：2019-02-13 14:11

【摘要】：自1980年以来,我国数学竞赛教育主要以全国高中数学联赛为载体,并取得长久而稳定的发展。在联赛中有一些重要的知识可谓是“年年必考、常考常新”,数列问题就包含在其中。长期以来,有关数列的试题一直都受到联赛命题组的青睐。本文试图以数学联赛为出发点进行内容研究,并聚焦到数列问题这一具体的考点上,通过对近16年所有数列问题的搜集整理,探寻规律,以对一线教师、参赛者以及命题者提供一些思路和参考。本文共分为五章,两大部分。首先对全国高中数学联赛的发展历程进行了回顾,包括其赛制及题型的变化,然后重点研究并总结了联赛的命题原则。第二部分基于第一部分的研究,并针对2000-2015年全国高中数学联赛中的“数列”试题进行了收集。首先给出等差和等比数列的定义及性质,和《新课标》下对于数列学习的要求作为前期铺垫。然后,从宏观与微观的角度对这些试题进行了研究。宏观上,通过表格的形式列举了近16年联赛中所有数列题的题号、题型、分值以及所考察的知识点。这些数据经过多方面多角度的统计,给出了相应的分析图表,并总结出数列问题在联赛中的一些发展特点。微观上,深入到每一道题的解题过程,对历年真题中较为典型的例题的解题思路进行了分类研究。第一类题型是由递推式求数列通项。在本文第四章第一节中介绍了解题中最常用的不动点法,特征根法,待定系数法和数学归纳法等。第二类题型是数列求和问题。在本文第四章第二节给出了解决此类问题的主要方法,包括化归法,裂项相消法和错位相减法。第三类是利用数列性质的解题。数列众多完善的性质让它在考试中作为考查点可以灵活多变.本文第四章第三节,给出了数列单调性、整除性、无穷性、有界性的定义,并把考查数列性质的历年真题进行了分类解答。第四类是一些综合性试题,涉及数列与其他知识点的综合运用。最后,结合以上研究所总结的规律及特点进行应用,给出了一个关于数学竞赛辅导之数列问题的教案,实践到课堂中去,以期为一线教育工作者提供一些教学的参考。
[Abstract]:Since 1980, the mathematics competition education of our country mainly takes the national senior high school mathematics league as the carrier, and obtains the long-term and steady development. In the league there are some important knowledge is "every year test, often test new", the number of problems included in it. For a long time, the number series questions have been favored by league proposition group. This article attempts to take the mathematics league as the starting point to carry on the content research, and focuses on the number series question this concrete examination spot, through to nearly 16 years all the number series question collection arrangement, seeks the rule, in order to the first line teacher, Participants and propositions provide some ideas and references. This paper is divided into five chapters, two parts. This paper reviews the development course of the national senior high school mathematics league, including the change of its competition system and question type, and then studies and summarizes the proposition principle of the league. The second part is based on the first part of the study, and the National High School Mathematics League from 2000 to 2015 in the "number series" questions collected. At first, the definition and properties of isodyne and equal ratio sequence are given, and the learning requirements of number series under "New Curriculum Standard" are used as the prelude. Then, from the macro and micro point of view, these questions are studied. Macroscopically, this paper enumerates the number, type, score and the knowledge points of all the number series problems in the last 16 years by tabular form. Through the statistics of many aspects and many angles, the corresponding analysis charts are given, and some development characteristics of the series problems in the league are summarized. From the micro point of view, deep into the process of solving each problem, the author classifies the ideas of solving some typical examples of real problems in the past years. The first type of problem is to find the general terms of the series from the recursion. In the first section of the fourth chapter, the most commonly used fixed point method, characteristic root method, undetermined coefficient method and mathematical induction method are introduced. The second type of problem is the summation of number series. In the second section of chapter 4, the main methods to solve this kind of problems are given, including the method of reduction, the method of disjunctive phase elimination and the method of dislocation subtraction. The third is the use of the nature of the number of the solution. The numerous and perfect nature of the number series makes it flexible and changeable as the examination point in the examination. In the third section of chapter 4, the definitions of monotonicity, divisibility, infinity and boundedness of sequence are given. The fourth category is a number of comprehensive test questions, involving the comprehensive use of numbers and other knowledge points. Finally, based on the application of the rules and characteristics summarized by the above research, a teaching plan about the number series of problems of mathematics competition guidance is given, which is put into practice in the classroom, in order to provide some teaching reference for the first-line educators.
【学位授予单位】：西北大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：G633.6

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