Unit 9 · Lesson 9.1

Arithmetic & geometric sequences as functions

A sequence is just an ordered list of numbers, like 3, 7, 11, 15, and so on. Nothing exotic. You make lists like this whenever you count by fives or read off a row of dates. What turns it into algebra is one question: what are you doing to get from each number to the next?

Here's a way to picture it that connects back to Unit 4. There, a function was a machine: you feed in a number, and out comes exactly one number. A sequence is that same machine, fed the counting numbers in order.

Feed in 1, out comes the 1st number on the list. Feed in 2, out comes the 2nd. The position in the list is the input; the number sitting there is the output. So a sequence really is a function. Its inputs just happen to be 1, 2, 3, and so on. We write the 1st term as a_1, the 2nd as a_2, and the term in position n as a_n.

What we're really after is the rule the machine is using. Look at 3, 7, 11, 15. What do you do to get from each term to the next? You add 4, every time. Now look at 2, 6, 18, 54. What's happening here? Each term is the one before times 3. Those are the two behaviors this lesson is built on, and they get names. Add the same number each step is an arithmetic sequence. Multiply by the same number each step is a geometric sequence.

Notice that you just said the rule out loud. "Each term is the one before, plus 4" is a complete recipe for building the list, as long as you also know where to start. That little phrase is what we'll call the recursive rule in a moment, and you understood it before any symbols showed up. The only catch is that it needs a starting number, or it has nowhere to begin.

When you're not sure which kind of sequence you're looking at, there's one small tool that settles it every time. Write the terms in a row, and underneath, write the jumps between them. Do this two ways: write the differences (what you'd add) and the ratios (what you'd multiply by). Whichever one stays the same tells you the type.

For the first list the differences are all +4, while the ratios aren't constant, so it's arithmetic. For the second the ratios are all ×3, so it's geometric. When a sequence has you guessing, compute both rows and look for the one that holds steady. That table is the whole classification tool.

There are two ways to write any of these sequences in symbols. They describe the same list; they just answer different questions.

The first is the step-by-step way, the one you already spoke aloud. It says how to get each term from the one right before it. For an arithmetic sequence with common difference d, that's a_n = a_{n−1} + d, and you have to state the first term a_1 separately so it has somewhere to start. For a geometric sequence with common ratio r, it's a_n = r·a_{n−1}, again with a_1 given. This is called the recursive rule, and it mirrors exactly what you do by hand: take the last number, do the one move, write the next number.

The second way skips the walk. Suppose you want the 100th term and you don't fancy listing 99 numbers first. The jump-ahead rule, called the explicit rule, takes you straight there from the term number alone. Here's where it comes from. To reach term n, you start at a_1 and take steps. But term 1 takes zero steps, because you're already there. Term 2 is one step out, term 3 is two steps out, so term n is n−1 steps out. That's the whole reason an n−1 shows up:

$$\text{arithmetic:}\quad a_n = a_{n-1} + d,\ a_1\text{ given}\qquad\Longleftrightarrow\qquad a_n = a_1 + (n-1)\,d$$ $$\text{geometric:}\quad a_n = r\cdot a_{n-1},\ a_1\text{ given}\qquad\Longleftrightarrow\qquad a_n = a_1\cdot r^{\,n-1}$$

Moving between the two forms is easier than it looks, because they share their parts. The d (or r) in the step-by-step rule is the same d (or r) in the jump-ahead rule, and a_1 is the same in both. So once you've read off the first term and the common difference or ratio, you can write either form.

One more thing, and it's the precise reason a sequence isn't quite the same as the line or curve it resembles. The inputs to a sequence are only the counting numbers: 1, 2, 3, and so on. There's no "term 1.5." You can't ask the machine for a number it has no button for.

So if you plot the terms of 3, 7, 11, 15 as points (1, 3), (2, 7), (3, 11), (4, 15), you get a row of separate dots, not a connected line. The dots happen to sit exactly on the line y = 4n − 1, but the sequence is only the dots. This is what we mean by a discrete domain: the inputs come in separate, countable steps with nothing in between.

The line that runs through the dots is a continuous parent that fills in the gaps; the sequence itself doesn't. An arithmetic sequence is a linear function looked at only on the counting numbers. A geometric sequence is an exponential function looked at only on those same whole-number inputs.