TABLE OF CONTENTS In this appendix a few useful definitions and results are gathered for reference.
Proof The result is easily proved by induction, which requires demonstrating the truth of the formula for n = 1 (which is obvious) and showing that if the formula is true for any positive integer n, then it must also be true for n + 1. This follows since if 
and we assume that S n = n( n + 1)/2, then necessarily
proving the claim.
The sum can also be expressed as
Proof This can also be proved by induction, but for practice we note another approach. Just as in solving differential or difference equations, one can guess a general form of solution and solve for unknowns. Since summing k up to n had a second-order solution in n, one might suspect that solving for a sum up to n of squares of k would have a third-order solution in n, that is, a solution of the form f( n) = an 3 + bn 2 + cn + d for some real numbers a, b, c, d. Assume for the moment that this is the case, then if f( n) = 
, clearly n 2 = f( n)
