The first part of this paper presents a method to calculate the sum of the same powers of the first n positive integers which is non-recursive and easy to express algorithmically. The application is demonstrated through several problems, for example by calculating the sum of arithmetic progression of degree p. The second part of the paper shows that the discussed procedure can also be used to calculate the Bernoulli numbers, and then, with the help of a known theorem, a link is established between the sum of the same powers of the first n positive integers and the Bernoulli numbers.