Research and Proof on the Summation Law of a Class of Unit Fraction Sequences with Triangular Number Denominators

In the study of sequence summation in elementary mathematics, the method of telescoping cancellation is the core approach for solving the summation of unit fraction sequences. Conventional solutions require term-by-term expansion of the general term, which is relatively cumbersome. This paper focuses on a class of unit fraction sequences with specific triangular number denominators. Through observation, induction, calculation and verification, it independently summarizes an extremely simple summation law for such sequences. Relying on the method of accumulation to revise the general term of the denominator sequence, the method of telescoping cancellation is used to complete a rigorous mathematical proof of the law. Multiple examples are combined to verify the correctness of the conclusion, and the applicable boundaries of the law are clearly defined. The revised derivation process in this paper is free of mathematical errors, which can provide a simple solution for the summation of similar sequences in elementary mathematics and also serve as a reference case for inquiry-based learning of sequences.