Generating functions are a powerful tool to investigate properties of sequences. We'll look at how they are used to solve simple recurrences, to give easy solutions to combinatorics problems, and to study asymptotic properties of sequences.

One of the crowning theorems of elementary number theory, the study of natural numbers, is quadratic reciprocity. It is a result about the remainders when you divide perfect squares by primes; it is among the most elegant I have ever seen. Fortunately, it also happens to be very accessible. We will sketch a proof of this theorem assuming no more than arithmetic.