对数列通项公式的探索与求解是数列中的典型问题，在高考中非常容易考出来，有的时候以小题（填空或选择）的形式出现，但绝大多数情况下是在大题中设一问求解数列通项。是数列问题的核心，是我们分析数列性质的重要依据， 也是学好数列知识的关键。其实总体来说，所有的数列问题在大部分情况下，就相当于是求它的通项公式，尤其是那些难度系数较大的数列问题，数列通项公式的探求往往是解数列的关键。本文主要从一般递推数列、一阶递推数列、二阶线性齐次、非齐次递推数列、 阶递推数列这几方面来探讨、研究它的通项公式的推导方法。论文研究的最终结果是熟练掌握数列通项公式的几种一般推导方法及了解一阶、二阶、 阶线性递推数列的通项公式的常用推导方法。数列递推关系是诸多重要数学思想方法的载体，作为离散函数的典型模型，既具备函数的性质，又具有自己独特的递推关系，与其它的知识更有着相当密切的联系,这些决定了它在研究数列性质的重要地位.而稳定性是数列的一个重要的性质之一,而从数列递推关系出发来研究数列的稳定性已经取得比较丰硕的成果,但是利用乘m數列共通性的方法来研究数列递推关系的稳定性却是非常少的,针对这一点,本文以乘m數列共通性这个新的角度来切入研究数列递推关系的稳定性.且本文针对一阶数列递推关系的稳定性做了乘m數列共通性的探究,结果发现探究的结果和严格证明的结果一致,且更能以直观的效果来判定其稳定性。本文的研究是直观方法的角度来研究问题的,这与传统的纯代数式证明方法有很大的不同点.其次,本文的研究为数列递推关系稳定性的严格证明提供了方向.另外,本文为读者提供了一种以新角度探索问题的学习方式.
关键词: 递推关系;稳定性;乘m數列共通性
Abstract
The exploration and solution of the general term formula of the sequence is a typical problem in the sequence, which is very easy to be found out in the college entrance examination, sometimes in the form of small questions (fill in the blanks or choose), but in most cases, the general term of the sequence is set in the big questions. Is the core of the problem of sequence, is an important basis for us to analyze the nature of sequence, is also the key to learn sequence knowledge. In fact, in general, all sequence problems in most cases, it is equivalent to finding the general term formula, especially those sequence problems with greater difficulty coefficient, the search for the general term formula is often the key to solving the sequence. This paper mainly from the general recursive sequence, the first order recursive sequence, the second order linear homogeneous, inhomogeneous recursive sequence, the order recursive sequence these aspects to explore and study its general term formula derivation method. The final result of this paper is to master several general derivation methods of the general term formula of sequence and to understand the common derivation methods of the general term formula of sequence of first order, second order and order linear recursion. Sequence recursive relation is the carrier of many important mathematical thinking methods. As a typical model of discrete function, it not only has the property of function, but also has its own unique recursive relation. More has a close connection with other knowledge, which determines its important position in the nature of research series. The stability is one of an important part of the nature of the sequence, and starting from the sequence of recursive relations to study the stability of the series has made more fruitful results, but used by m sequence convergence of the method to study the stability of the recursive relations is very little, for this, this article take the m series convergence of the new Angle to cut into the study of the stability of the series recursive relations. And the stability of the first-order series recursive relations, this paper made by m series convergence of exploration, the results showed that the result of the inquiry and strict The results of Ming dynasty are consistent, and the stability of Ming dynasty can be judged by its intuitive effect. The research in this paper is to study the problem from the perspective of intuitive method, which is quite different from the traditional pure algebraic proof method.
Key words: recursive relation; Stability; Multiplication by m is common
1. 引言
1.1 研究背景
在乘m數列共通性方法的指导下，应用其研究成果对数列递推关系的稳定性研究方法进行反思，提出新的解题思路和方法，并在今后的数列递推关系的问题探讨实现以下目的：
对于递推关系稳定性的研究,也就是递推关系极限的探讨,目前有许多已经通过严格证明的方法:郑华盛的《非线性递推数列极限的不动点解法》[1]利用不动点理论解决了非线性分式和二次函数递推数列极限的求法,利用不动点思想的角度来求证递推数列的稳定性.梁志清的《用不动点定理讨论递推数列的收剑性》[2]利用不动点定理讨论由递推关系χm+1=f(χn) 给出的数列{χn∞}的收敛性并探讨递推数列的收敛性与函数之间的关系.吴延东的《递推数列χm+1=f(χn)的单调性与收敛性讨论》[3]利用函数单调递增对递推数列χm+1=f(χn)单调性进行讨论。
1.2 本文主要研究的问题
本文通过对递推关系和数列的相关研究,探究如何利用乘m數列共通性的方法有效的应用在求证递推关系的稳定性.具体主要研究以下四个方面的问题：探究哪些数列递推关系适用于用乘m數列共通性的方法来论证其稳定性重点说明怎样利用乘m數列共通性的方法来探究数列递推关系的稳定性通过分析比较,探讨乘m數列共通性方法和常用方法分别研究数列稳定性的关系对数列递推关系的思维进行开拓，在乘m數列共通性方法指导下的高中数列递推关系的探讨与前瞻。
1.3 研究的意义
培养创新的思维意识和扩宽探究问题的角度.数学问题的求解有时候是殊途同归的,探究的方法会更加新颖,掌握了这样的能力,对我们以后探究任何的问题都是大有裨益的.
6 结论
乘m數列共通性的方法为解决数列递推关系稳定的问题提供非常便利的途径,综合对比用乘m數列共通性方法解决一阶线性常系数递推关系稳定性问题和一阶非线性常系数递推关系稳定性问题时,可以发现对于复杂性较低的数列递推关系,用乘m數列共通性的方法来解决并没有体现它的优势,但对于较为复杂的数列递推关系,用乘m數列共通性的方法就大大简化了题目的求证过程,甚至可以一步到位得出数列的稳定性,所以说,乘m數列共通性的方法的优势是显而易见的.不过,乘m數列共通性的方法也有明显的缺点,那就是它可适用的数列递推关系比较局限,有比较固定的递推形,例如对于高阶的递推关系就不能用此方法来解决。
参考文献（略）
