Analysis of solutions to competitive problems on the maximum and minimum of sequences in high school
高中数列最值竞赛题解法例析

摘要

数列最值问题是高中数学竞赛的核心考点之一，其解法融合函数单调性、不等式性质、递推逻辑等多模块知识，对学生的综合应用能力要求较高。本文针对竞赛题的命题特点，基于递推关系法、函数法、不等式法、构造法等多种典型解法，结合2022-2024年全国联赛、各省预赛及历年经典真题共8道例题，详细拆解解题逻辑与步骤，兼顾方法普适性与竞赛题的灵活性，为竞赛备考提供系统性参考。

Abstract

The problem of finding the extreme values of sequences is one of the core topics in high school mathematics competitions. Its solutions integrate knowledge from multiple areas, including the monotonicity of functions, properties of inequalities, and recursive reasoning, demanding a high level of comprehensive application from students. This article, focusing on the characteristics of competition problems, explores various typical methods such as the recursive relation method, function method, inequality method, and construction method. By analyzing eight example problems from the 2022-2024 National League, provincial preliminaries, and classic historical problems, it provides a detailed breakdown of the problem-solving logic and steps. It balances the general applicability of methods with the flexibility required for competition problems, offering a systematic reference for competition preparation.