| _version_ | 1866901528213716992 |
|---|---|
| author | lihong |
| author_facet | lihong |
| contents | <p>In junior high school mathematics competitions and high school entrance examination finale problems, there exists a special type of fractional equation whose denominators form an arithmetic sequence and the values of each fraction also form an arithmetic sequence. Traditional common denominator simplification methods are cumbersome and error-prone for such equations, and there is a risk of extraneous roots due to the non-equivalent deformation in the process of eliminating denominators. This paper proposes a mathematical model of “double arithmetic sequence isomorphic fractional equations”, strictly derives its compatibility conditions, general rapid solution formulas and right-hand constant value formulas, and establishes an infinitely expandable question construction system. The system requires no common denominator operation, directly solves integer solutions, and transforms root verification into simple arithmetic sequence verification, fundamentally avoiding the trap of extraneous roots. Meanwhile, this paper conducts methodological sublimation from the perspectives of isomorphism thought and mathematical mechanization. The model not only serves exam-oriented education but also helps students develop structured thinking.</p> <p>在初中数学竞赛及中考压轴题中,存在一类分母成等差数列且分式取值亦成等差数列的特殊分式方程。针对此类方程,传统通分化简方法计算繁琐、易错,且因去分母过程的非同解变形存在增根风险。本文提出“双等差同构分式方程”的数学模型,严格推导其相容性条件、通用速解公式以及右端定值公式,并建立可无限拓展的题型构造体系。该体系无需通分,可直接求解整数解,将“验根”转化为简单的等差数列验证,从根本上规避增根陷阱。同时,本文从同构思想、数学机械化等角度进行了方法论升华,使模型不仅服务于应试,更助力学生结构化思维的形成。</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20222508 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Structured Solution and Question Construction System of Double Arithmetic Sequence Isomorphic Fractional Equations lihong <p>In junior high school mathematics competitions and high school entrance examination finale problems, there exists a special type of fractional equation whose denominators form an arithmetic sequence and the values of each fraction also form an arithmetic sequence. Traditional common denominator simplification methods are cumbersome and error-prone for such equations, and there is a risk of extraneous roots due to the non-equivalent deformation in the process of eliminating denominators. This paper proposes a mathematical model of “double arithmetic sequence isomorphic fractional equations”, strictly derives its compatibility conditions, general rapid solution formulas and right-hand constant value formulas, and establishes an infinitely expandable question construction system. The system requires no common denominator operation, directly solves integer solutions, and transforms root verification into simple arithmetic sequence verification, fundamentally avoiding the trap of extraneous roots. Meanwhile, this paper conducts methodological sublimation from the perspectives of isomorphism thought and mathematical mechanization. The model not only serves exam-oriented education but also helps students develop structured thinking.</p> <p>在初中数学竞赛及中考压轴题中,存在一类分母成等差数列且分式取值亦成等差数列的特殊分式方程。针对此类方程,传统通分化简方法计算繁琐、易错,且因去分母过程的非同解变形存在增根风险。本文提出“双等差同构分式方程”的数学模型,严格推导其相容性条件、通用速解公式以及右端定值公式,并建立可无限拓展的题型构造体系。该体系无需通分,可直接求解整数解,将“验根”转化为简单的等差数列验证,从根本上规避增根陷阱。同时,本文从同构思想、数学机械化等角度进行了方法论升华,使模型不仅服务于应试,更助力学生结构化思维的形成。</p> |
| title | Structured Solution and Question Construction System of Double Arithmetic Sequence Isomorphic Fractional Equations |
| url | https://doi.org/10.5281/zenodo.20222508 |