ABSTRACT
Sums of powers of positive integers have been of interest to mathematicians since antiquity. Over the years, mathematicians in various places have given verbal formulas for the sum of the first n positive integers, the sum of the squares of the first n positive integers, the sum of the cubes of the first n positive integers, and so on. Beginning as early as the tenth or eleventh century, general methods existed. However, since each sum depended on the sums of the lower powers and required extensive new calculation, often done entirely verbally, in practice, these general methods did not result in calculation of formulas for sums of very high powers.This research article mainly focuses on the derivation of generalized result of this sum. More explicit formula has been derived in order to get the sum of any arbitrary integral powers of first n-natural numbers. Furthermore by using the fundamental principles of Combinatorics and Linear Algebra an attempt has been made to answer an interesting question namely: Is the sum of integral powers of natural numbers a unique polynomial? As a result it is depicted that this sum always equals a unique polynomial over natural numbers. Moreover some properties of the coefficients of this polynomial are derived.More importantly a recurrence relation which can give the formulas for sum of any positive integral powers of first n-natural numbers has been proposed and it is strongly believed that this recurrence relation is the most significant thing in this entire discussion
