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 
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 
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 
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 
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.
Mean Green Math 
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