In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them. Today’s quip is one that I’ll use when simple techniques get used in a complicated way.
Consider the solution of the linear recurrence relation
where and . With no modesty, I call this one the Quintanilla sequence when I teach my students — the forgotten little brother of the Fibonacci sequence.
To find the solution of this linear recurrence relation, the standard technique — which is a pretty long procedure — is to first solve the characteristic equation, from , we obtain the characteristic equation
This can be solved by any standard technique at a student’s disposal. If necessary, the quadratic equation can be used. However, for this one, the left-hand side simply factors:
(Indeed, I “developed” the Quintanilla equation on purpose, for pedagogical reasons, because its characteristic equation has two fairly simple roots — unlike the characteristic equation for the Fibonacci sequence.)
From these two roots, we can write down the general solution for the linear recurrence relation:
where and are constants to be determined. To find these constants, we plug in :
To find these constants, we plug in :
We then plug in :
Using the initial conditions gives
This is a system of two equations in two unknowns, which can then be solved using any standard technique at the student’s disposal. Students should quickly find that and , so that
which is the final answer.
Although this is a long procedure, the key steps are actually first taught in Algebra I: solving a quadratic equation and solving a system of two linear equations in two unknowns. So here’s my one-liner to describe this procedure:
This is just an algebra problem on steroids.
Yes, it’s only high school algebra, but used in a creative way that isn’t ordinarily taught when students first learn algebra.
I’ll use this “on steroids” line in any class when a simple technique is used in an unusual — and usually laborious — way to solve a new problem at the post-secondary level.