浅谈构造法在数列中的运用

四川师范大学本科毕业论文

浅谈构造法在数列中的运用

学生姓名 院系名称 专业名称 班 级 学 号 指导教师 完成时间

数学与软件科学学院 数学与应用数学

浅谈构造法在数列中的运用

学生姓名： 指导教师：

内容摘要:构造法，就是根据题设条件或结论所具有的特征、性质，构造出满

足条件或结论的数学模型，借助于该数学模型解决数学问题的方法。利用利用构造法求数列的通项公式是高考中的常考点之一，解题思路比较简单、可操作性强。但是利用构造法求数列的前n项和的可操作性则较弱。本文就是通过举例来说明构造法在数列求通项公式和前n项和中的一些运用，并简要说明一些通过构造数列的方法来证明一些不等式题型的方法。

关键词：构造法 数列 不等式

How to Apply the Construction Method in Sequence

Abstract:Construction method, is a way of which is based on the

characteristics of the hypothesis or conclusion to build a mathematical model which is constructed to meet the condition and conclusion, with which to solve mathematics problems.The general term formula for the sequence which is constructed by using construction method is often one of the examination points in the college entrance examination. With this method, the way of problem-solving is relatively simple and strong operability. But for the sum of the first n terms of the sequence which is constructed by using construction method is weak in its maneuverability.This article is through the way of giving examples to illustrate some application of construction method for general term formula in sequence and the sum of the first n terms, and is a brief description of some ways by constructing a sequence to prove some inequality questions.

Key words:Construction Sequence Inequality

目录

1引言............................................................1

2构造法在数列求通项公式中的运用..................................2 2.1直接构造一个等差数列或等比数列............................2

2.2形如an?1?pan?f(n)（p为常数，且p?0,p?1）的数列........2 2.3形如“pan?2?qan?1?ran”型的数列...........................4 2.4用特征方程构造等差数列或等比数列..........................6 2.5取倒数构造等差数列或等比数列..............................6 2.6取对数构造新的等差或等比数列..............................7 2.7公式变形构造..............................................7

2.8通过换元来构造新的数列求解................................8

2.9对于两个数列的复合问题，也可构造等差或等比数列求解........9 2.10其他特殊数列的特殊构造方法...............................9 3构造法在数列求和中的运用........................................11 3.1逐差构造法................................................11

3.2利用组合数公式构造数列的通项求和..........................12 4构造数列证明不等式..............................................12 4.1直接法....................................................13

4.2作差法....................................................13 4.3作商法....................................................14

4.4差分法....................................................15 4.5商分法....................................................15 5总结............................................................16 参考文献...........................................................17 致谢...............................................................18