The general term hn of a sequence is a polynomial in n of...

The general term $h_{n}$ of a sequence is a polynomial in $n$ of degree $3 .$ If the first four entries of the Oth row of its difference table are $1,-1,3,10$, determine $h_{n}$ and a formula for $\sum_{k=0}^{n} h_{k}$.

Key Concepts

Difference Tables

A difference table is a structured way of computing and organizing the successive differences of the terms in a sequence. Each row of the table represents a level of differences, and for a polynomial of degree d, the (d+1)-th row will consist of all zeros, with the d-th row being constant. This method is particularly useful in solving for the polynomial when only a few initial terms or differences are known.

Summation of Polynomial Sequences

Summing a polynomial sequence involves finding an expression for the cumulative sum of its terms. Since the sum of a polynomial of degree d results in another polynomial, typically of degree d+1, standard formulas or techniques such as telescoping sums, Faulhaber's formula, or direct summation of each term's contribution are used. This concept is essential for analyzing series and computing their partial sums in closed form.

Polynomial Sequences

A polynomial sequence is one in which each term is given by a polynomial function of its index. When the degree of the polynomial is known (in this case, degree 3), it implies that the general term can be expressed in the form a?n³ + a?n² + a?n + a?. Understanding polynomial sequences is crucial for determining their behavior, such as growth rates, and for finding explicit expressions for the terms and their sums.

Finite Differences

Finite differences involve computing the differences between successive terms of a sequence. For a polynomial sequence of degree d, the d-th finite differences are constant. This property is used to identify the degree of the polynomial and assists in determining the coefficients of the polynomial that defines the sequence.

Transcript

Please give Ace some feedback