I am attempting to solve the problems in chapter 1 - Basic Properties of Numbers from Spivak's book.
Problem 1(v) asks the student to prove that
x^n - y^n = (x - y)(x^(n - 1) + (x^(n - 2))y + ... + x(y^(n - 2)) + y^(n - 1))
The Attempt at a Solution
I solved it by using induction, but induction is not taught until later in the book.
I assume you can solve the exercises solely using the material in the previous chapters, or the hints in the exercise.
After repeatedly applying P9 (distributive law of multiplication) it is obvious that all the terms cancel, except for the first and the last, but proofs with elipses (..) don't strike me as rigorous. So, how can you prove this without using induction?