A simple mnemonic to compute sums of powers

TL;DR

The paper introduces a simple recursive method for computing sums of powers of natural numbers, avoiding complex constants like Bernoulli numbers, making it accessible for educational and competitive contexts.

Contribution

It presents a novel, easy-to-remember recursive formula for sums of powers that does not require advanced constants, simplifying calculations especially in educational settings.

Findings

Provides a recursive formula for sums of powers

Eliminates need for Bernoulli numbers in calculations

Suitable for use in mathematics competitions

Abstract

We give a simple recursive formula to obtain the general sum of the first $N$ natural numbers to the $r$th power. Our method allows one to obtain the general formula for the $(r+1)$th power once one knows the general formula for the $r$th power. The method is very simple to remember owing to an analogy with differentiation and integration. Unlike previously known methods, no knowledge of additional specific constants (such as the Bernoulli numbers) is needed. This makes it particularly suitable for applications in cases when one cannot consult external references, for example mathematics competitions.