双等差数列的性质及其应用

On the Properties and Applications of Bi-Arithmetic Sequences

摘要: 双等差数列作为一类具有交替变化特性的特殊数列，在理论研究和实际应用中具有重要意义。本文系统研究了双等差数列的性质：首先证明了数列构成双等差数列的充要条件是其奇数项子列与偶数项子列均为等差数列且公差相同；其次建立了按固定间隔提取子列时的等差或双等差特性，并研究了前n项和序列的相应性质。本文还引入了双等差数列的生成函数，分析不同公差关系下数列的渐近行为与周期性。最后，通过精密激光切割设备工作台位移建模等实例，展示双等差数列在工程应用中的独特价值。

Abstract: Bi-arithmetic sequences, as a special class of sequences with alternating variation characteristics, hold significant importance in both theoretical research and practical applications. This paper systematically investigates the fundamental properties of bi-arithmetic sequences. We first prove that a necessary and sufficient condition for a sequence to be bi-arithmetic is that both its odd-term and even-term subsequences are arithmetic sequences with a common difference. In addition, we establish the arithmetic or bi-arithmetic characteristics of subsequences extracted at fixed intervals, and study the corresponding properties of partial sum sequences. We also introduce the generating function of bi-arithmetic sequences and analyze their asymptotic behavior and periodicity under different common difference relationships.. Finally, through examples such as displacement modeling of precision laser cutting equipment worktables, we demonstrate the unique value of bi-arithmetic sequences in engineering applications.