A classic type of problem that students of all grade levels are asked is “What’s the next number in this pattern?”, where the pattern is an arithmetic sequence
$$5, 8, 11, 14, 17, \ldots $$

If you’ve worked with second and third graders, they’ll take a little bit of time, but eventually they see that the difference between successive numbers is \(3\). Although, they’ll never explain it that way. They may say something like “add by three”.

Students in higher grade levels tend to pick up on the constant difference scheme more quickly than their lower grade level counterparts, but their explanation doesn’t really change. Nor does this explanation really change when students start taking Algebra and some formalities of arithmetic sequences are introduced. Prior to an Algebra class, this type of pattern is often simply called “Patterns” and problems usually fall into the bucket of “fun math” or “extracurricular math”.

There is also the snarky response that you’ll get from someone who is a little too advanced for their own good. That snarky response usually goes something like this, “Well, the next number in the pattern is any number I want it to be because I can fit an arbitrary degree polynomial to those numbers. These are stupid problems!” Technically, this is correct, but also misguided. If we are going to be fitting a form to data, then in the above example, \(y = 3x + 2\) fits perfectly (\(R^{2} = 1\)) and is the lowest degree polynomial that does it. Hence, of all the models one can choose from, \(y = 3x + 2\) is the best as it involves the fewest number of parameters and attains a perfect fit.

Anyway …

The point of this article is to talk about arithmetic sequences and how their instruction in an Algebra class is often botched. Typically, arithmetic sequences show up in an Algebra course as part of the introduction to sequences, series, and recursion. Students are given a progression of numbers like that above and are shown that this progression can be written as $$a_{n+1} = a_{n} + d$$ From here, students are asked, “Well, what is \(a_{n}\)?” and are shown that $$a_{n} = a_{n-1} + d$$ and hence, that $$a_{n+1} = a_{n-1} + 2d$$ by substitution — another Algebra-learned technique. From here, some hand-waving at the induction process, students are “convinced” that $$a_{n+1} = a_{1} + nd$$ or in textbook standard form, $$a_{n} = a_{1} + (n-1)d$$

Bleh.

All of this is technically correct and completely not understood by Algebra students. Here are a handful of reasons why:

• There is a cognitive overload with the subscript notation. Students are barely used to function notation, and suddenly this notation with subscripts and (effectively) functions within subscripts is introduced.
• As if \(a_{n}\) and \(a_{n+1}\) weren’t bad enough, there is also \(n-1\) not in the subscript. Overload extreme!!
• In addition to the sequence notation overload, there is the general symbolic and jargon overload: \(d\) is called the common difference, \(a_{1}\) is the initial term and often is special in that it is supposed to be a given.

While I understand that this is now an Algebra class and one of the points is to get the hang of symbolic manipulation, the lack of a conceptual understanding of the problem is a huge hindrance to understanding symbolic manipulation. It’s not even that “Hey those kids are just moving symbols around without knowing what they’re doing”, it’s more that “those kids don’t know how to move the symbols around because they have no idea what the underlying problem is”. And this brings me to the title of this article and my appeal to “Times Table Sequences”.

How I Teach Arithmetic Sequences

Let’s start with the problem from above …

Find the next several terms in $$5, 8, 11, 14, 17, \ldots $$

Easy since we are just adding by threes, we have \(20, 23, 26, \ldots\).

At this point, I pause the problem for a moment and ask students if they can just count by threes starting at three. Of course, they can and they recite \(3, 6, 9, 12, 15, \ldots\). They can also recognize that this is the threes times table (or if they don’t immediately recognize it, a simple, leading question does the trick).

Now, we are ready! I stack the two sequences in question on top of each other like so

$$\begin{matrix}
5 & 8 & 11 & 14 & 17 \\
3 & 6 & 9 & 12 & 15
\end{matrix}$$

What additional pattern do they notice when comparing the stacked sequences? “The top sequence is always two greater than the bottom sequence.” or in Algebra student speak “It’s just two more than the other one.” — and I encourage instructors to clean up this ambiguous language.

From here, I ask if they can tell me what the \(5000^{th}\) term in the top sequence is. Sometimes it takes a little coaxing, but students can come up with the reasoning that the \(5000^{th}\) term is \(3\times 5000 + 2\). And there we have it — the top sequence is just the threes times table with an offset of two.

Pretty much every student I have had in a College Algebra class has seen arithmetic sequences in high school. And pretty much every one of these students doesn’t get it. Once, they can see that this sequence is nothing more than a times table sequence, the entire topic runs on its own.

Standard questions that involve negative \(d\) become easy [hey it’s just a negative times table!]. Questions that give the \(n^{th}\) and \(m^{th}\) aren’t too much of a challenge [it takes a little thinking, but Algebra students figure out that a division is necessary — and if you’re feeling ambitious you can relate it back to finding the slope of a line]. Once they’ve done a whole bunch of these problems without the symbolism of sequences, then introducing the sequence notation is far more easy. Students know where the problem is headed.

I strongly discourage this overly aggressive practice of immediately introducing new symbolism. This is “simple but evil” in that the topic is relatively easy for students to grasp [they already know times tables], it’s evil in that the standard approach is unhelpful.