探究斐波那契数列的通项公式及简单性质

To Explore the General Term Formula and Simple Properties of Fibonacci Sequence

摘要: 本文介绍了人教A版《普通高中课程标准实验教科书•选择性必修二》中斐波那契数列的呈现内容，为解决如何从递推公式推导出斐波那契数列的通项公式，采用在数学归纳法的基础上证明通项公式，从结论出发分别采用构造等比数列、通过找特解和通解以及矩阵的特征值和特征向量的方式分别探求出斐波那契数列的通项公式，并根据研究数列的一般方法进一步探究斐波那契数列的简单性质，为开展数学探究活动提供范式。

Abstract: This paper introduces the presentation content of Fibonacci sequence in the Curriculum Standard Experiment Textbook for General Senior High School • Selective Compulsory II published in Human Education A edition. In order to solve the problem of how to derive the general term formula of Fibonacci sequence from the recursion formula, the general term formula is proved on the basis of mathematical induction. In the conclusion, the general term formula of Fibonacci sequence is explored by constructing equal ratio sequence, finding specific and general solutions and eigenvalues and eigenvectors of matrix, and the simple properties of Fibonacci sequence are further explored according to the general method of studying sequence, which provides a paradigm for carrying out mathematical inquiry.