A Constructive Method for Computing Sums of Powers

This paper presents a constructive method for computing sums of powers of consecutive integers, developed independently in 1980-1981 by the author at age 16-17. The method provides explicit formulas for calculating sums of the form: S_k(n) = 1^k + 2^k + 3^k + ... + n^k through an iterative approach based on iterated summations and Pascal's triangle coefficients. Three main methods are presented:• Method E₁: Direct computation via iterated summations• Method F₁: Recursive formulation using binomial coefficients • General Method: Unified framework connecting both approaches The methods produce closed-form expressions involving binomial coefficients and products of linear factors, verified numerically for k = 1, 2, 3, 4, 5. Unlike the standard approach using Bernoulli numbers or Faulhaber's formulas, this work employs a purely constructive technique that builds each sum from the previous ones through explicit algebraic operations. While the problem itself is classical, the particular approach and formulation appear to be original. This manuscript represents early mathematical work developed independently in 1980-1981, recovered from notebooks and formalized in 2026. It is published primarily for historical documentation and to preserve an independent derivation that may offer pedagogical value.