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Use Gauss's method to find a formula for the sum of the first n natural numbers.
Hint: The numbers of dots in the triangles are half of the numbers of dots in the following rectangles:
Assuming that f2(n) is a polynomial in n of degree 3, find this polynomial by interpolation.
Hint: Take (xi, yi) = (i, f2(i)) for each i = 1, 2, 3.
Prove that the degree-3 polynomial for f2(n) holds for all n.
Hint: For the base case, prove that the polynomial is 1 = f2(1) for n = 1. For the induction step, assume that the polynomial gives f2(n) for n and show that it gives f2(n + 1) for n + 1.
What is the difference fp – 1(n) – fp – 1(n – 1)?
Hint: What is the sum of the first n kth powers without the first n – 1 kth powers?
Since the pth Bernoulli polynomial is the unique polynomial (up to a constant) satisfying its definition, find a relationship of the function p fp – 1(n) to a Bernoulli polynomial (assuming that fp – 1(n) is a polynomial in n of degree p).
Hint: Since p fp – 1(n – 1) also satisfies the definition of Bp(n), the two must be equal up to an additive constant Cp.
Find the constant term of the polynomial fp(n).
Hint: Let n = 0 in the relationship of p fp – 1(n) to Bp(n).
Prove the conjectural formula for fp(n).
Hint: For each fixed p, use induction on n. The base case is that the polynomial is 0 at n = 0. For the inductive step, make use of the definition of Bp(n).