What is the Sum of Squares and General Polynomial Series?

The sum of squares and general polynomial series is a fundamental concept in mathematics, applicable in various fields from theoretical mathematics to real-world applications like computer science and physics. In this article, we will delve into the formula derived for the sum of squares of the first n natural numbers and extend the discussion to the sum of the first n elements of the sequence x^k, where n is a positive integer and k is an arbitrary degree polynomial.

Sum of Squares of the First n Natural Numbers

To find the n-th term of the series 1^2, 1^2 2^2, 1^2 2^2 3^2, ..., we identify the pattern in the terms. The n-th term of the series can be expressed as the sum of the squares of the first n natural numbers, denoted by T_n:

T_n 1^2 2^2 3^2 ... n^2

There is a well-known formula for the sum of the squares of the first n natural numbers:

T_n n(n 1)(2n 1) / 6

Therefore, the n-th term of the series is:

T_n n(n 1)(2n 1) / 6

This formula provides a concise and efficient method to calculate the sum of squares of the first n natural numbers for any positive integer n.

General Approach for Sum of Polynomials

For a more generalized approach, we consider the sum of the first n elements of the sequence px, where x is an arbitrary polynomial of degree k. This problem can be approached systematically by understanding the degree of the resulting polynomial sum. Specifically:

If the degree of the polynomial p is k, then the sum is a polynomial of degree k-1. A polynomial of degree k-1 is determined by k of its values.

Let's use this approach to find the sum of the first n elements of the sequence x^k.

Deriving the Formula

We know that the sum is a polynomial of degree k-1. Thus, we can write it in the form:

S_n a_{k-1}x^{k-1} a_{k-2}x^{k-2} ... a_1x a_0

To find the coefficients, we use the values for the first few terms:

S_0 0 S_1 a_{k-1} a_{k-2} ... a_1 a_0 1^k S_2 2^{k-1}a_{k-1} 2^{k-2}a_{k-2} ... 2a_1 a_0 1^k 2^k S_3 3^{k-1}a_{k-1} 3^{k-2}a_{k-2} ... 3a_1 a_0 1^k 2^k 3^k

For k2, we can solve the system of linear equations to find the coefficients:

0 a_0 1 a_1 a_0 1^2 4a_1 2a_0 1^2 2^2 5 9a_1 3a_0 1^2 2^2 3^2 14

Solving these equations, we find:

a_0 0 a_1 1/6 a_2 1/2 a_3 1/3

Thus, the formula for the sum of the first n elements of the sequence x^2 is:

S_n n^3/3 n^2/2 n/6

Proof by Mathematical Induction

To prove that the sum of the first n values of a polynomial p of degree k is a polynomial of degree k-1, we use mathematical induction:

Base Case: If k 0, then p is a constant c, so the sum is cn, which is a polynomial of degree 1 in n. Inductive Step: Assume the statement is true for every polynomial of degree k-1. If p is a polynomial of degree k, then the sum f_n sum_{j1}^{n} pj is a sum of polynomials of degree k-1. Since the sum of polynomials of degree k-1 is a polynomial of degree at most k-1, f_n is a polynomial of degree k-1.

This proof shows that the degree of the polynomial sum is preserved, making our approach valid for all polynomials.