Application of linear algebra in the study of sum of positive integral powers of first m-natural numbers

This research article explores on the summation of fixed positive integral powers of first m-positive integers. Authors are strongly believing that the following discussion depicts a method fromwhich one can easily compute the summation of fixed positive integral powers of first m-natural numbers. Furthermore they have given answers to two interesting questions in the research field of analytical number theory namely: Is the sum of fixed positive integral powers of first m-positive integers coincide with a polynomial?and Is such polynomial unique?. Some principles of linear algebra have been incorporated in this article to extract very interesting results regarding this summation. The uniqueness of the polynomial has been proved using Crammer’s rule also. Moreover a system of linear non homogeneous equations proposed here which help us to derive the formulas for the summations Σ n, Σ n2 , Σ n3…….