毕业论文数学归纳法的应用.doc

SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesTSelectionParbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbagraphFoLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointse11111111111111111111111111111111lectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraphFormatLineSpaci2222222222222222222222ngLinesToPoints2SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraphFccccccccccccccccccccccccccccccccccccccccccccccccccccccccormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraSelec 学学 士士 学学 位位 论论 文文 BACHELOR S THESIS SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesTSelectionParbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbagraphFoLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointse11111111111111111111111111111111lectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraphFormatLineSpaci2222222222222222222222ngLinesToPoints2SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraphFccccccccccccccccccccccccccccccccccccccccccccccccccccccccormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraSelec 编号编号 学学士士学学位位论论文文 数学归纳法的应用数学归纳法的应用 学生姓名 凡 丽 学 号 20040102025 系 部 数 学 系 专 业 数学与应用数学 年 级 04 2 班 指导教师 王 卫 东 完成日期 2008 年 5 月 1 日 SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesTSelectionParbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbagraphFoLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointse11111111111111111111111111111111lectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraphFormatLineSpaci2222222222222222222222ngLinesToPoints2SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraphFccccccccccccccccccccccccccccccccccccccccccccccccccccccccormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraSelec 学学 士士 学学 位位 论论 文文 BACHELOR S THESIS SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesTSelectionParbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbagraphFoLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointse11111111111111111111111111111111lectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraphFormatLineSpaci2222222222222222222222ngLinesToPoints2SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraphFccccccccccccccccccccccccccccccccccccccccccccccccccccccccormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraSelec 1 中文摘要 数学归纳法是中学数学中一种常用的证题方法 它的应用极其广泛 本文讨 论了数学归纳法的步骤 它集归纳 猜想 证明于一体 体现了数学归纳法的证 题思路 本文归纳总结了数学归纳法解决代数恒等式 几何 排列组合等方面的 一些应用问题的方法 并对应用中常见的误区加以剖析 以及一些证法技巧介绍 有利于提高对数学归纳法的应用能力 关键词 关键词 数学归纳法 步骤 SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesTSelectionParbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbagraphFoLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointse11111111111111111111111111111111lectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraphFormatLineSpaci2222222222222222222222ngLinesToPoints2SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraphFccccccccccccccccccccccccccccccccccccccccccccccccccccccccormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraSelec 学学 士士 学学 位位 论论 文文 BACHELOR S THESIS SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesTSelectionParbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbagraphFoLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointse11111111111111111111111111111111lectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraphFormatLineSpaci2222222222222222222222ngLinesToPoints2SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraphFccccccccccccccccccccccccccccccccccccccccccccccccccccccccormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraSelec 2 AbstractAbstract Mathematical induction is a common evidence method in secondary school mathematics it is have very broad application In this paper author reaserch into the step of the Mathematical induction it includes summariz evidence and guess embody the idea of the evidence of mathematical induction Also at here we summariz themethod of the mathematical induction application in solve algebra identities geometric order and portfolio and so on also analyze the common errors on application and into duct skill of the proof proof of skills introduced It is help to increased the level of the Mathematical induction s application KeyKey wordswords Mathematical induction steps SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesTSelectionParbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbagraphFoLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointse11111111111111111111111111111111lectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraphFormatLineSpaci2222222222222222222222ngLinesToPoints2SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraphFccccccccccccccccccccccccccccccccccccccccccccccccccccccccormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraSelec 学学 士士 学学 位位 论论 文文 BACHELOR S THESIS SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesTSelectionParbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbagraphFoLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointse11111111111111111111111111111111lectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraphFormatLineSpaci2222222222222222222222ngLinesToPoints2SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraphFccccccccccccccccccccccccccccccccccccccccccccccccccccccccormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraSelec 3 目 录 中文摘要中文摘要 1 1 ABSTRACTABSTRACT 1 1 引言引言 2 2 1 1 数学归纳法的历史由来数学归纳法的历史由来 1 1 1 1 基本步骤 2 1 2 基本形式 3 3 数学归纳法解决应用问题 数学归纳法解决应用问题 4 4 2 1 代数恒等式方面的问题 4 2 2 几何方面的应用 4 2 3 排列和组合 5 2 4 对于不等式的证明 有时适当放大或缩小 有时用综合法和分析法 6 6 3 3 常见误区及剖析常见误区及剖析 7 7 3 1 忽略了归纳的基础 7 3 2 归纳推理出错 7 7 SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesTSelectionParbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbagraphFoLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointse11111111111111111111111111111111lectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraphFormatLineSpaci2222222222222222222222ngLinesToPoints2SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraphFccccccccccccccccccccccccccccccccccccccccccccccccccccccccormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraSelec 学学 士士 学学 位位 论论 文文 BACHELOR S THESIS SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesTSelectionParbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbagraphFoLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointse11111111111111111111111111111111lectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraphFormatLineSpaci2222222222222222222222ngLinesToPoints2SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraphFccccccccccccccccccccccccccccccccccccccccccccccccccccccccormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraSelec 4 4 4 应用技巧应用技巧 9 9 4 1 配凑 归纳假设 法 9 4 2 分析法 1010 5 5 数学归纳法的推广数学归纳法的推广 1111 6 6 小结小结 1212 参考文献参考文献 1313 致谢致谢 1414 SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesTSelectionParbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbagraphFoLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointse11111111111111111111111111111111lectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraphFormatLineSpaci2222222222222222222222ngLinesToPoints2SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraphFccccccccccccccccccccccccccccccccccccccccccccccccccccccccormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraSelec 学学 士士 学学 位位 论论 文文 BACHELOR S THESIS SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesTSelectionParbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbagraphFoLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointse11111111111111111111111111111111lectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraphFormatLineSpaci2222222222222222222222ngLinesToPoints2SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraphFccccccccccccccccccccccccccccccccccccccccccccccccccccccccormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraSelec 1 SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesTSelectionParbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbagraphFoLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointse11111111111111111111111111111111lectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraphFormatLineSpaci2222222222222222222222ngLinesToPoints2SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraphFccccccccccccccccccccccccccccccccccccccccccccccccccccccccormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraSelec 学学 士士 学学 位位 论论 文文 BACHELOR S THESIS SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesTSelectionParbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbagraphFoLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointse11111111111111111111111111111111lectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraphFormatLineSpaci2222222222222222222222ngLinesToPoints2SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraphFccccccccccccccccccccccccccccccccccccccccccccccccccccccccormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraSelec 2 引 言 在中学数学学习的过程中 有一种很常见且基本的数学方法 数学归纳 法 对于数学归纳法 有人问 为什么说数学归纳法是严格的证明方法 数学归 纳法的原理是什么 数学归纳法的证明过程为什么要有这样的规定格式 数学归 纳法的应用前景如何 下面将逐一进行解答 1 数学归纳法的历史由来 曾经有一个叫皮亚诺的意大利人把我们小时侯数数的过程归纳整理出来 称 作正整数公理 这个公理有五条 简单归纳一下 前四条是说 1 是正整数 且它不是任何正整数的后面的一个数 称作后继 即 1 是第一个正整数 每个 正整数都有唯一的后继 而且是正整数 关键是第五条 一个正整数集合 如果包含 1 并且假设包含 也一定包含它的后继 那这个集合包含所有的正x 整数 这一条就是数学归纳法的原理 用符号表示 即 1 如果 且满足 2 若则 那么 SN S 11Sk 1kS SN 根据这一原理 就有了数学归纳法 设是与正整数有关的命题 如果 P n 当时正确 即正确 1 1n 1 P 若假设正确前提下 可以证明命题也正确 2 P k 1 P k 那么命题对任意正整数都是正确的 数学归纳法的正确性可以用 正整数最小数原理 加以证明 正整数最小数 原理是说 任何非空正整数集合一定含有最小数 SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesTSelectionParbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbagraphFoLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointse11111111111111111111111111111111lectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraphFormatLineSpaci2222222222222222222222ngLinesToPoints2SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraphFccccccccccccccccccccccccccccccccccccccccccccccccccccccccormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraSelec 学学 士士 学学 位位 论论 文文 BACHELOR S THESIS SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesTSelectionParbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbagraphFoLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointse11111111111111111111111111111111lectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraphFormatLineSpaci2222222222222222222222ngLinesToPoints2SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraphFccccccccccccccccccccccccccccccccccccccccccccccccccccccccormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraSelec 3 1 1 基本步骤 数学归纳法是数学中一种重要而独特的证明方法 对与自然数有关的命题n 的证明是行之有效的 首先它的两个步骤缺一不可 其次它的应用非常广泛 可以用它解决好多方面的数学问题 数学归纳法的步骤 2 当时 这个命题是正确的 1 1n 假设当时 这个命题是正确的 那么当时 这个命题也是正 2 nk 1nk 确的 数学归纳法的两个步骤缺一不可 一方面不要认为 一个命题在的时候1 n 正确 在时正确 在时也正确 则这个命题就正确了 老实说 不要2n 3n 说当的时候正确不算数 就是为 1000 的时候正确 或者 1 万的时候正确 3n n 对任何自然数是否正确 还得证明了再说 1 2 基本形式 常见的数学归纳法有下列几种 第一数学归纳法 假设是一个与自然数有关的命题 若正确 P n 1 1 P 假设成立 可推出 成立 则对一切自然数成立 2 P k 1 P k P n 第二数学归纳法 若是一个关于自然数的命题 若正确 假 P nn 1 1 P 2 设对成立 可推出成立 则对一切自然数都成立 P nkn 1 P k P n 跳跃归纳法 设是一个关于自然数的命题 若对 1 2 3 P nn 1 P nn j 成立 假设成立 可推出成立 则对一切自然数 2 P k jkP P n 都成立 n 双重归纳法 设是一个关于两个独立自然数有关的命题 若 P m n n m 1 SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesTSelectionParbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbagraphFoLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointse11111111111111111111111111111111lectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraphFormatLineSpaci2222222222222222222222ngLinesToPoints2SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraphFccccccccccccccccccccccccccccccccccccccccccccccccccccccccormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraSelec 学学 士士 学学 位位 论论 文文 BACHELOR S THESIS SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesTSelectionParbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbagraphFoLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointse11111111111111111111111111111111lectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraphFormatLineSpaci2222222222222222222222ngLinesToPoints2SelectionParagraphFormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPointselectionParagraphFccccccccccccccccccccccccccccccccccccccccccccccccccccccccormatLineSpacingLinesToPointsSelectionParagraphFormatLineSpacingLinesToPoctionParagraSelec 4 成立 假设成立 可推出与成立 1 1 P 2 P m n 1 P mn mnP 1 则对一切自然数成立 3 P m n m n 数学归纳法解决应用问题数学归纳法解决应用问题 通过对数学归纳法的了解 我们不难发现 它的应用是十分广泛的 用这 个数学方法可解决以下几个方面的数学问题 4 2 1 代数恒等式方面的问题 有不少的代数恒等式 它的严格证明需要用到数学归纳法 例1 数列的第项 可以用公式 表示 这里是它的首n 1 1 n aand