Topic 2.1: Change in Arithmetic and Geometric Sequences

Five millimeters

This little domino is five millimeters tall, about the size of a grain of rice. That’s a penny lying next to it, just so you can see how tiny it really is. Right behind it stands another domino, one and a half times bigger in every direction, and behind that one another, one and a half times bigger again, and the line just keeps going down this road.

Now, watch closely. I’ll give the little one a gentle push. A domino can knock over another one about one and a half times its size, so every fall sets off a slightly bigger fall, all the way down the line. A physicist at the University of Toronto really built a chain like this, thirteen dominoes long, and the last one weighed about a hundred pounds.

If you kept going, somewhere down this road there’d be a domino taller than the Empire State Building. By the end of this lesson, you’ll be able to tell me exactly which domino that is. And you’ll know how tall every single domino before it is, without measuring any of them.

Sixty pennies

Your brain is going to fight you on this, because multiplying by the same number over and over feels a lot like adding. That little penny is the most famous proof I know.

Imagine a penny that doubles every day. On day one you have one cent. On day two you have two cents, and on day three you have four. So, just go with your gut. How much do you think that penny is worth on day thirty?

Sixty cents is such a reasonable mistake to make, because if you add two cents every day you crawl up two, four, six, eight, sixteen cents on day eight, only thirty by day fifteen, and sixty cents on day thirty. But multiply by two every day and you go one, two, four, eight, and you end up a millionaire five times over, and that difference between adding and multiplying is what this whole lesson is about.

No domino two and a half

Let’s go back to the road. This time, look at the first eight dominoes from the side. I’ll number the dominoes starting at zero, because the number counts how many steps along the chain you are. The first domino is zero steps along.

So each domino now has its own spot, one for every whole number. Now put a dot on top of each domino. The number underneath is the input, and the height of its top is the output. Watch the dots land on paper.

And that’s exactly what a sequence is, a function whose inputs are the whole numbers, zero, one, two, three and so on, so its graph is a set of separate dots. You might want to join them with a smooth curve, but there’s no domino two and a half, nothing standing halfway between domino two and domino three, so the graph stays as separate dots.

We name the heights with a subscript, so g sub zero is five millimeters, g sub one is seven and a half, g sub two is eleven and a quarter, and the subscript is the input, the domino’s number. Notice that the first domino is g sub zero, not g sub one, because it’s zero steps along the chain. Hang on to that. It’s going to try to trip you up later.

Slide or zoom

Now let’s build a second chain in the next lane, starting with the same two dominoes, five and seven and a half millimeters. But after that, each one is just two and a half millimeters taller than the last.

Let’s ride along its top, one domino at a time. Each time, I move the camera one domino over and two and a half millimeters up, and look, the picture doesn’t change at all. Only the number on the road changes, because this chain adds the same amount every step, which makes it an arithmetic sequence, and you can slide along it and it looks the same everywhere.

The original chain would grow right out of the frame if I slid along it. So instead, here’s the trick. Every time a domino falls, I’ll pull the camera back one and a half times.

Watch the scale bar, five millimeters, seven and a half, eleven and a quarter. Every fall looks the same, the same shot, only bigger, and that bar is the only thing that changes. This chain multiplies by the same factor every step, so it’s a geometric sequence, and you can zoom along it and it looks the same at every scale.

Here are both chains as tables, with the heights first, and they start out exactly alike, five and then seven and a half, but at the third domino the arithmetic chain has ten while the geometric one already has eleven and a quarter. And by the fourth domino, it’s twelve and a half against almost seventeen. For the arithmetic chain, subtract each height from the next, so seven and a half minus five is two and a half, and every step after that is two and a half again.

That steady step is called the common difference, d, and it’s really a constant rate of change, two and a half millimeters per domino. For the geometric chain, subtracting gives a new number every time, so we divide instead, and seven and a half over five is one and a half, just like every step after it.

That’s the common ratio, r, and it means every domino is fifty percent bigger than the one before it, a constant proportional change. So to tell them apart, subtract the neighbors and divide the neighbors. Whichever one keeps giving the same number tells you what you’ve got. Let’s try one that has nothing to do with dominoes. What kind of sequence is this one?

Equal steps, bigger steps

Let’s look closely at how much each domino adds. On the arithmetic chain, the slice sticking up above the domino before it is always the same, two and a half millimeters. On the geometric chain the first slice is two and a half millimeters too, but the next is three point seven five. That’s where the two chains part ways.

So how much taller is the fourth domino than the third? I’ve written both heights for you. And it keeps going, eight point four, twelve point seven, nineteen, so the geometric chain’s risers are multiplying by one and a half themselves, while the arithmetic chain’s never change. So an increasing arithmetic sequence goes up by the same amount every step, while an increasing geometric sequence of positive values goes up by more every step, and that’s the kind of sentence the AP exam wants you to write from a table.

And the word positive matters, because negative numbers like minus eight, minus four, minus two, minus one make a geometric sequence that goes up too, but by less and less every step. By the eleventh domino, the geometric chain is almost a foot tall, and the arithmetic one is barely more than an inch.

Domino number n

So how do we reach a domino far down the road without multiplying by one and a half dozens of times? Let’s build a shortcut, starting with the arithmetic chain. The first domino is a sub zero. To reach the next one you add d once, to reach the one after you add d twice, then three times, so to reach domino n you add d n times, and a sub n equals a sub zero plus d times n.

For our arithmetic chain, that’s five plus two point five n millimeters. The geometric chain works the same way with multiplying. One step multiplies by r, two steps by r twice, r squared, and n steps by r to the n, so g sub n equals g sub zero times r to the n.

Adding d again and again became d times n. Multiplying by r again and again became r to the n. So the height of domino n is five times one point five to the n millimeters.

And here the zero comes back. The exponent counts the steps between dominoes, and from the first domino to the tenth there are only nine, like a fence with ten posts and nine gaps, so the tenth domino is g sub nine.

Let’s use it. Say you’re about one point seven meters tall, five foot seven, so tap the first domino in the line that’s taller than you.

Start anywhere

Sometimes you don’t even know where a sequence starts. Say the only domino you’ve measured is the tenth, and it’s nineteen point two centimeters, about as tall as a pencil. To reach the twentieth we don’t need the first one at all, we just hop, one, two, three, four, five, six, seven, eight, nine, ten, each hop times one and a half, and we land on a domino about eleven meters tall, as tall as a three-story house.

So from any domino k you know, you reach domino n by multiplying by r once per hop, and there are n minus k hops, so g sub n equals g sub k times r to the n minus k. For us, g sub nineteen is g sub nine times one point five to the tenth, and it’s ten hops whether you number the dominoes from zero or from one.

Arithmetic works the same way. Add d each hop, so a sub n is a sub k plus d times n minus k. You can hop backward. Then n minus k is negative, and each hop back divides by r or subtracts d. g sub seven is two hops before g sub nine, so divide by one and a half twice. That’s about eight and a half centimeters.

Here’s a real one. The Summer Olympics are numbered, and Paris twenty twenty-four was officially the Games of the thirty-third Olympiad, with a new Olympiad starting every four years. So let’s jump ahead to the year on the board. Which Olympiad will that be?

Two terms are enough

This next one is exactly how the exam likes to ask it. A geometric sequence has g sub two equals twelve and g sub five equals ninety-six, and we want g sub n. From term two to term five is three hops, so ninety-six has to be twelve times r cubed. Divide both sides by twelve and r cubed is eight, and the number that cubes to eight is two, so r is two.

Starting from the term we know, that’s g sub n equals twelve times two to the n minus two, and that’s a perfectly good answer. Walking back two hops to g sub zero, twelve divided by two twice is three, so the very same sequence is three times two to the n. Two different-looking formulas, one sequence, and the exam will happily give you the answer in whichever form you didn’t write.

Now you try one the way the exam asks it. Here’s a geometric sequence with positive terms. Which expression gives its nth term?

Twelve frets

Dominoes are a sequence somebody built on purpose, but there’s one hiding on every guitar you’ve ever seen. Look at the frets, the metal strips across the neck, and notice how they crowd closer together as you go up. That isn’t for looks, because when every fret has to raise the note by the same musical step, the frets can’t be spaced any other way.

Press a string at a fret and only the part between that fret and the bridge vibrates, and every fret shortens that part by the same factor, about zero point nine four four. So the lengths go twenty-five and a half inches, about twenty-four, twenty-two point seven, a geometric sequence with a ratio less than one.

Twelve frets of that leave exactly half the string. That’s why the twelfth fret sits right in the middle, usually with two dots on it. The pitch goes the other way. A shorter string vibrates faster.

The open A string plays a hundred ten hertz, a hundred ten vibrations a second. Each fret multiplies that by about one point oh six. Twelve frets up, you’ve doubled it to two hundred twenty. Twelve more, at the very last fret, doubles it again to four hundred forty hertz, the A that musicians all over the world tune to. Watch the string land on paper.

Four forty is the note every tuner starts from. It’s our known term, like the tenth domino. Now g sub n is the pitch at fret n. Twelve frets double the pitch, so r to the twelfth is two. That makes r about one point oh six.

Start from four forty at fret twenty-four, and g sub n is four forty times r to the n minus twenty-four. Your turn. Start from four forty at the very last fret. What pitch plays at fret twelve?

Middle C on a piano is fret fifteen, nine frets below fret twenty-four. Its pitch is four forty times r to the negative nine, about two hundred sixty-one point six hertz.

The twenty-ninth domino

Let’s go back to the desert. It’s getting dark. And we still owe ourselves an answer. The Empire State Building’s roof is three hundred eighty-one meters up, that’s three hundred eighty-one thousand millimeters, and domino n is five times one point five to the n millimeters tall. So, counting the little one as number one, which domino in the line is the first one taller than the roof of the Empire State Building?

Come on. Let’s fly over there. We’ll see it up close. Four hundred twenty-six meters, taller than the roof and just short of the tip of the antenna, and it all started with a domino the size of a grain of rice, multiplied by one and a half only twenty-eight times. And the chain that added two and a half millimeters every time has a twenty-ninth domino seventy-five millimeters tall, which you could hold in your hand.

Let’s put the whole lesson on one board. A sequence is a function of the whole numbers, so its graph is a set of separate dots. An arithmetic sequence adds the same amount, d, every step, so a sub n is a sub zero plus d n, or a sub k plus d times n minus k from any term you know.

A geometric sequence multiplies by the same factor, r, every step, so g sub n is g sub zero times r to the n, or g sub k times r to the n minus k. An increasing arithmetic sequence climbs by the same amount every step, and an increasing geometric one with positive values climbs by more every step. Whichever one you’re working with, count the steps, not the terms.

So the next time you see a line of dominoes, you’ll know you can name the height of any one of them from a single one you’ve measured, and you’ll know exactly which one would stand taller than the Empire State Building.