I’m using the Twelve Days of Christmas (and perhaps a few extra days besides) to do something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series concerning one of my favorite lectures concerning various applications of geometric series.
Part 1: Introduction to generating functions.
Part 2: Enumeration problems; or counting how many ways $2.00 can be formed using pennies, nickels, dimes, and quarters. (The answer is 1463.)
Part 3: The generating function for the Fibonacci sequence.
Part 4: Using a generating function to find a closed-form expression for the (ahem) Quintanilla sequence, a close but somewhat less famous relative of the Fibonacci sequence.
Part 5: Reproving the formula for the Quintanilla sequence using mathematical induction.