1. The problem statement, all variables and given/known data Use the method of Problem 6 to show that ∑1≤k≤n kp can always be written in the form (np+1) / (p+1) +Anp+Bnp-1+Cnp-2+... Source: Calculus by Michael Spivak. Chapter 2 problem 7. 2. Relevant equations The method from problem 6 is described as follows: The formula for the sum of consecutive squares may be derived as follows. We begin with the formula (k+1)3 - k3=3k2+3k+1 Writing this formula for k=1,..,n and summing all n equations we arrive at (n+1)3 -1=3(sum of consecutive squares)+3(sum of consecutive positive integers)+n or (n+1)3 -1=3(12+22+...+n2)+3(1+2+...+n)+n 3. The attempt at a solution I have included (uploaded) all of my work. It's too much to write here and every step is not super relevant. In short I thought that the only way to go about this was to do an inductive proof. After a lot of algebra I have arrived at the following equation found at the end of Page4. ∑rk=1 kp+1= (rp+2) / (P+2) +rp+1 -1/(P+2)[∑0≤j≤p∑1≤k≤r-1(p+2 C j)(kj) +1] Which almost what I need. That last term is the collection of all r with exponents ≤p but I can't figure out how to expand it so I have coefficients that I can relabel as A, B, C, etc. Thanks for the help.