Years and years ago, back in my old Math League days, I came up with a trick to finding out the formulas of sequences. The problem type I'm talking about would give me a list of Xs and Ys in a table like this:
Wait, I need HTML for tables? Wow, okay.
| X | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Y | 4 | 8 | 14 | 22 | 32 |
Oh my god, I thought I hated tables when I only knew of how awful they were in the context of word processors. HTML tables are so much worse.
Anyways, the problems would have a table like the one above and ask for the equation that generated those values, usually in the form of y = f(x). There aren't any convenient algorithms for this, so usually the student is expected to graph it, determine if it's a line or parabola, and use the appropriate equations.
This problem is actually one of the few interesting math problems that a student might see in school. I categorize a lot of math taught in schools as a three step process:
1. Remember the formula or algorithm that applies to this problem.
2. Apply the formula or algorithm and follow the steps.
3. Write down the result.
This problem gets an additional step; an analysis of the results, as simple as it may be.
1. Remember the formula or algorithm that applies to this problem (Hm, when the problem looks like this, I have to graph the points on a Cartesian plane)
2. Apply the formula or algorithm and follow the steps. (Actually graph each point)
3. Analyze the results. (Check if the graphed points follow a line or a parabola)
4. Remember the formula or algorithm that applies to this problem (Remember the appropriate formulas for your graph)
5. Apply the formula or algorithm and follow the steps. (Use the appropriate formulas / plug 'n chug)
6. Write down the result (The formulas spit out this number for the slope and this number for the y-intercept, so I can put them in place of the m and the b and write down the equation)
Yeah, that additional analysis step is very simple, but it's one of two examples that come to mind in the math I learned as a kid that weren't completely algorithmic. (The other is triangle congruence, which was always a favorite of mine!)
This analysis step is something I loved back then, because I could get around it with some thinking. Why bother graphing out all those points when you can just intuitively figure out whether the points will make a line or a parabola? (Well, because the teacher's going to take points away for not showing your work, that's why.)
So, how do you figure out whether it's a line or a parabola without graphing it? I came up with a method that I'm now going to retroactively call "Sequence Differences". I called them "subsequences" back then, but it turns out that that term is protected. I subtracted adjacent y values from each other to make a new sequence that had one fewer term. Let's see if I can get a table to illustrate that:
| X | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Y1 | 4 | 8 | 14 | 22 | 32 |
| Y2 | 4 | 6 | 8 | 10 |
Eh, close enough. As you can see, the Y2 sequence comes from 8 - 4, 14 - 8, 22 - 14, and 32 - 22. Upon seeing this, I thought "these numbers are all different, and lines have a constant slope, so it must be a parabola!" and went on to the next part of the problem. But isn't it interesting that the numbers are increasing in an arithmetic sequence? If I were to do another round of sequence differences here,
| X | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Y1 | 4 | 8 | 14 | 22 | 32 |
| Y2 | 4 | 6 | 8 | 10 | |
| Y3 | 2 | 2 | 2 |
we can see that the sequence is constant. Somehow, every time I performed a sequence difference, I decremented the degree of the source equation. The first equation was y = x2 + x +2; the second can be solved to be y = 2x + 2, and the last is just y = 2. Looks... a bit familiar, doesn't it? The constant isn't the same, but it looks almost like we're taking the derivative to find the equations that describe our new sequences. There's a mystery there!
But, that's probably plenty for one day; it's definitely enough HTML tables for one day. All that remains is a name, and I think I've got a good one. Differential Equations are often abbreviated to Diff Eq, pronounced "Diffy Q", and this problem is about Differences (of) Sequences, so let's go with the almost cringeworthy Diffsy Q. Next time, a deep dive into the mechanics of Diffsy Qs!
