Abstract

This paper investigates a part of Faulhaber formula studied in the curriculum. Specifically, it examines the formula for sum_{k=1}^n k^{alpha} (alpha = 1, 2, 3). Based on textbook analysis, this study attempts to derive a general method of inductive justification for the formula when α=1, 2, 3), and further extends it to the cases where α=4, 5. First, through an analysis of previous research, two consistent methods of inductive justification were identified. One is the 'geometric rotational symmetry' method, and the other uses ratios. Second, usingthese two methods, the formula for sum_{k=1}^n k^{alpha} was inductively derived for α=4, 5.