数学四年级上册重难点及概念

周口师范学院本科毕业论文

题目：例谈数列递推关系及通项公式的求解方法

系别：数学与与信息科学系

专业：数学与应用数学

班级： 05专升本

学号：200504020025

姓名：张艳蕊

导师姓名：刘学忠

2007年05月20日

例谈数列递推关系及通项公式的求解方法

摘要

数列是高中教材的重点内容,也是高考考查的重要知识点,我们知道数列内容考查的重点是数列的通项,特别是由递推关系给出的数列求通项公式,它在高级中学数学教学中是一个难点,又占有一定的重要位置,其题目也是灵活多变,而且数列关系的给出也是多种多样:直接给出递推关系式,以

a与n s为背景间接给出递推关系式,以方程为背景

n

间接呈现递推关系式,以不等式为背景给出递推关系式。针对这些类型的数列递推关系式,将已知数列递推关系式用代数的一些变型技巧整理变形。尽管求递推数列通项公式的方法也非常灵活,但是,只要善于把握递推关系式的的特征,往往可以通过适当的策略问题化归为等差数列,等比数列问题加以解决,亦可采用完全归纳的方法,由特殊情形推导出一般情形,进而用数学归纳法加以证明。以下通过典型例题介绍了十二种已知数列的初始条件和递推方程,求通项公式的一般性方法.它对由数列递推关系式求通项公式提供了可以尝试的途径。对数列通项公式的求解方法归纳整理为:迭代法,逐差累加法,逐商累乘法,换元法,积差相消法,取倒数法,取对数法,平方(开方)法,待定系数法,猜想法,不动点法,特征根法。

关键词: 数列递推式累加法累乘法

Talking about relation between series and general formula

of solution

ABSTRACT

Series ,as the important part of high-school curriculum, is also the key knowledge point which the collage entrance examination check .as we know, the essential point that the series examines is the General Formula, especially getting it through series given by recursive relationship, Which is a difficult part and plays an important role in senior teaching, whose subjects are also very flexible. In addition, it offers us in various way, such as, given directly, given indirectly according to the relationship between General Formula and Sn, occurring indirectly by way of equation or given through inequality, For all these types of Series recursive relations, we should sort them out with some algebra techniques. The methods of getting General Formula is very flexible, but we can always transform it into arithmetic series by appropriate strategies only if we master the characteristics.

When we deal with the geometric series, we can also adopt the complete—induction, Which deduce generalization from specialization, then verify it with mathematical induction.

The following gives us 12 kinds of general methods by tipical examples to get General Formula according to the known initial condition and the recursive equation. They are :iteration, the cumulative PV poor, tired gradually to multiplication, for yuan, According to the plot differential phase, for the countdown, for the few, square (prescription), to be determined coefficient, conjecture, fixed point, Eigenvalues law. Key words : Series Recursive CK Tired multiplication

目录

中文摘要 (Ⅰ)

英文摘要 (Ⅱ)

1迭代法 (5)

2逐差累加法 (6)

3逐项累乘法 (6)

4换元法（构造数列法） (7)

5积差相消法 (9)

6取倒数法 (10)

7取对数法 (10)

8开方（平方）法 (11)

9待定系数法 (11)

10不动点法 (13)

11特征根法 (14)

12猜想法（例举归纳法） (16)

结束语 (16)

参考文献 (18)

致谢 (19)