An Engineer's Approach to a General Algorithm for Finding the Sum of Powers of Natural Numbers

Abstract

Convenient formulae for finding the sums of kth power of the first n natural numbers may be useful in some engineering applications. The formulae for k = 1, 2, and 3, are commonly found in the literature. In this paper, an attempt is made to develop a general algorithm for finding the sum for any positive integer value of k. The development of the algorithm is entirely an engineering approach, based purely on a simple geometric interpretation and does not involve any deep mathematics. This algorithm may be used to derive the formulae for different values of k. Some of the possible engineering applications of these formulae are also discussed.

A Simple Tool for Engineers for Generating Formulae for the Sum of Powers of Natural Numbers

“…It has been shown that the sum of the kth power of the first n natural numbers may be expressed as a polynomial of order (k + 1) without a constant term [1]. Hence, it may be expressed as: (1) where a k,j is the coefficient of n j of the polynomial.…”

“…Hence, it may be expressed as: (1) where a k,j is the coefficient of n j of the polynomial. The formulae presented in reference [1], together with some more formulae, may be expressed in polynomial form and the values of various coefficients may be tabulated as in Table 1. It may be noted that k = 0 is an obvious result which is included only to complete the triangular form of Table 1.…”

“…It has been shown that the sum of the kth power of the first n natural numbers may be expressed as a polynomial of order (k + 1) without a constant term [1]. Hence, it may be expressed as: (1) where a k,j is the coefficient of n j of the polynomial. The formulae presented in reference [1], together with some more formulae, may be expressed in polynomial form and the values of various coefficients may be tabulated as in Table 1.…”

“…An algorithm for finding the sum of the kth power of the first n natural numbers for k ≥ 3, has been developed [1]. The development was based on purely a geometric interpretation.…”

“…The development was based on purely a geometric interpretation. Using this algorithm, formulae for different values of k have been derived and presented, and some possible applications of these formulae in the field of engineering have been discussed [1]. The main aim of that paper was to inculcate among young and budding engineers essential engineering aptitudes, such as creativity, and geometric and physical interpretation.…”