PrepShorts · Study sheet · Class 6 Mathematics · Chapter 1, Patterns in MathematicsPrepShorts

Chapter 1 · Patterns in Mathematics

Every number sequence is a rule, not a list

यह वीडियो हिंदी में भी · Watch in Hindi

Number sequences12 min

This video could not be loaded. Reload the page to try again.

Sign in with Google

12 min.

Also recorded in Hindi.Englishहिन्दी

Two rows of the same printed table both start 1, 2, 3 — and then one goes to 4 and the other to 5. No run of numbers can tell you which sequence you are looking at.

The idea

A row of Table 1 shows a handful of numbers — seven on nine of the ten rows, six on the cubes — but the sequence is not those numbers. It is the machine that made them, and what is printed is only what the machine got through before the page ran out. The proof is inside the table itself: two different rows open with 1, 2, 3 and then part company, so no run of digits can identify a sequence. That is why the book asks for the rule in the student's own words and not only for three more numbers.

What you should be able to do

  • Distinguish a sequence from a finite list of its early values
  • Read each row of Table 1 and state its rule in the student's own words
  • Produce the next three values of any row of Table 1 and justify them from the rule rather than by eye
  • Show that a shared opening does not determine a sequence, using two rows of Table 1 that begin the same way
  • Classify Table 1's rules by how each one gets from a value to the next: a fixed step, a growing step, a multiplier, or a look-back
  • Name the whole numbers as the objects §1.2 says these patterns are made of, and number theory as the study of them
  • State where the book's Virahānka sequence begins, which is not where most other books start it

Words to know

TermDefinition in one lineFirst introduced
number sequencean unending run of numbers produced by one ruleprinted in §1.2, p.2
whole numbersthe counting numbers together with 0printed in §1.2, p.2
number theorythe study of whole-number patternsprinted in §1.2, p.2
rulethe instruction that produces the next value from what came beforeprinted in the §1.2 Figure it Out, p.3
counting numbers1, 2, 3, 4, … — the row of Table 1 that starts at 1printed in Table 1, p.3
odd numbers1, 3, 5, 7, …printed in Table 1, p.3
even numbers2, 4, 6, 8, …printed in Table 1, p.3
triangular numbers1, 3, 6, 10, … — each value adds one more than the last additionprinted in Table 1, p.3
squares1, 4, 9, 16, … — a number multiplied by itselfprinted in Table 1, p.3
cubes1, 8, 27, 64, … — a number multiplied by itself twice overprinted in Table 1, p.3
Virahānka numbers1, 2, 3, 5, 8, … — each value is the sum of the two before itprinted in Table 1, p.3
powers of 21, 2, 4, 8, … — repeated doublingprinted in Table 1, p.3
powers of 31, 3, 9, 27, … — repeated triplingprinted in Table 1, p.3
term (of a sequence)one value at one position in a sequencean added word; not printed in this chapter

Where people slip up

  • "If I can continue it, I know it." Refuted from inside Table 1: after 1, 2, 3 the honest answer is which row? Continuing is evidence for a rule, never a substitute for one.
  • "Every sequence goes up by a fixed amount." Only three of Table 1's ten do: the counting numbers, the odd numbers and the even numbers. The All 1's row does not go up at all. Of the remaining six, three change their step every time — the triangular numbers, the squares and the cubes — and two multiply instead of adding. The sixth, the Virahānka row, does neither: its steps run 1, 1, 2, 3, 5, 8, so the step does not change every time, and nothing is being multiplied. It looks back instead.
  • "The Virahānka numbers start 1, 1, 2, 3, 5." This book prints them starting 1, 2, 3, 5, 8, 13, 21. The rule is the same — add the previous two — but the starting values differ from the version students meet online, and the examinable row is the printed one.
  • "Powers of 2 means 2, 4, 8, 16." The printed row begins at 1. So does the powers of 3 row. Anything multiplied by itself no times is 1, which is why.
  • "Squares and cubes are just names." §1.3 will cash both names out as pictures two pages later; treat the names here as promises, not decoration.
  • "Whole numbers and counting numbers are the same list." §1.2 starts the whole numbers at 0; Table 1's counting numbers start at 1.

Worked answers to this chapter’s exercises · this video explains Figure it Out · 1.2 Q1, Figure it Out · 1.2 Q2

Transcript1,594 words

One. Two. Three. Quick — say the next number out loud. Right now, before I do. You said four. Almost everybody says four, and it's a perfectly good answer. It's also not the only one. There is a table in your textbook, on page three, with ten rows of numbers in it. Two of those rows start one, two, three. One of them goes on four. The other one goes on five.

Same three numbers. Same page. Different sequences. So here's the question this whole video is about. If three numbers can't tell you which sequence you're looking at, what can? Here's the idea, and it's the one thing worth taking away today. A number sequence is not the numbers you can see. It's the rule that made them. Think of a rule as a little machine. You give it what you've got so far, and it hands you the next value.

What gets printed on the page is just whatever the machine managed to produce before the page ran out. In your book's table, most rows show seven values. One row, the cubes, shows only six. That's not a fact about the cubes. That's a fact about how much room the printer had. The sequence itself never stops. It goes on for ever, because the machine never gets tired. Before we meet the machines, let's be clear about what they're made of.

Your book says these patterns are patterns in the whole numbers. And it writes them out like this. Zero, one, two, three, four, and on for ever. Notice where that starts. Zero. The whole numbers include zero. Now, one of the rows in the table is called the counting numbers, and that row starts at one. So the whole numbers and the counting numbers are not the same list. They differ by exactly one number, and that number is zero.

It's a small difference and it's easy to skate over, so I'm saying it slowly. Whole numbers from zero. Counting numbers from one. And the study of patterns in these numbers has a name, which your book gives you in a single line. Number theory. Right. Here is the table, all ten rows, exactly as your book prints them. All ones. Counting numbers. Odd numbers. Even numbers. Triangular numbers. Squares. Cubes. Virahanka numbers. Powers of two. Powers of three.

Ten names, ten machines. And they are not ten unrelated things. Watch what happens if we sort them by how each one gets from one value to the next. Some take a fixed step every time. Some take a step that grows. Some don't add at all — they multiply. And one of them does something stranger, which we'll save for last. Four kinds of machine, ten rows. Let's take them one kind at a time.

Start with the simplest kind. A fixed step. All ones. One, one, one, one. The step is zero — it never changes. It's a machine that refuses to do anything, and it's still a rule. Counting numbers. One, two, three, four, five, six, seven. Add one, every time. So what comes next? Eight, nine, ten. And notice I didn't look those up. I applied the rule. Odd numbers. One, three, five, seven, nine, eleven, thirteen. Add two, starting from one.

Next three: fifteen, seventeen, nineteen. Even numbers. Two, four, six, eight, ten, twelve, fourteen. Add two again — but starting from two. Next three: sixteen, eighteen, twenty. Odd and even have exactly the same step. The only thing that separates them is where they begin. That's worth noticing. A rule isn't just the step. It's the step and the starting point. Second kind. A step that grows. Triangular numbers. One, three, six, ten, fifteen, twenty-one, twenty-eight.

Look at the jumps between them. Two, then three, then four, then five, then six, then seven. The steps are the counting numbers. Each time, you add one more than you added last time. So the next jump is eight, which takes us to thirty-six. Then nine, to forty-five. Then ten, to fifty-five. Now the squares. One, four, nine, sixteen, twenty-five, thirty-six, forty-nine. Their jumps are three, five, seven, nine, eleven, thirteen.

Which are the odd numbers. The squares grow by odd numbers. That's not a coincidence and section one point three will show you why, with a picture. The next jump is fifteen, giving sixty-four. Then seventeen, giving eighty-one. Then nineteen, giving one hundred. And the cubes. One, eight, twenty-seven, sixty-four, one hundred and twenty-five, two hundred and sixteen. This is the row that only prints six values, remember. Next come three hundred and forty-three, five hundred and twelve, and seven hundred and twenty-nine.

Third kind. Machines that don't add at all. Powers of two. One, two, four, eight, sixteen, thirty-two, sixty-four. Every value is double the one before. Not plus something — times two. So the next three are one hundred and twenty-eight, two hundred and fifty-six, and five hundred and twelve. Powers of three does the same thing with three. One, three, nine, twenty-seven, eighty-one, two hundred and forty-three, seven hundred and twenty-nine.

Next: two thousand one hundred and eighty-seven. Then six thousand five hundred and sixty-one. Then nineteen thousand six hundred and eighty-three. These get enormous, fast. That's what multiplying does. And here's the thing people get wrong about both rows. They start at one, not at two and three. Which looks odd until you say it out loud. Before you have doubled anything at all, you have one. Now the fourth kind. The strange one.

The Virahanka numbers. Your book prints them as one, two, three, five, eight, thirteen, twenty-one. Is the step fixed? One, one, two, three, five, eight. No. Is it growing steadily, like the triangular numbers? Not really — it starts with two ones. Is anything being multiplied? Two to three isn't doubling. Three to five isn't either. So this machine isn't doing any of the three things we've seen. It's looking back.

Each value is the sum of the two before it. Three and five make eight. Five and eight make thirteen. Eight and thirteen make twenty-one. So the next three are thirty-four, then fifty-five, then eighty-nine. One warning, and it matters for your exam. You may have met this sequence online starting one, one, two, three, five. This book does not print it that way. It starts one, two, three, five. Same rule, different starting values, and the printed row is the one you're being asked about.

Two of those names are doing something the others aren't. Squares. Cubes. Those aren't descriptions of the arithmetic. They're shapes. Nothing about adding odd numbers tells you why the word square belongs there. Your book is making you a promise. Two pages later, in section one point three, it draws these rows as pictures. And the shape turns out to be the reason. That's the pattern-and-reason idea from the last video, arriving right on schedule.

So hold on to those two names. There's a picture coming for both of them. Here's the task your book actually sets, on page three. Copy a row into your notebook. Write the next three numbers. Then write the rule in your own words. And it carries the Math Talk badge, which means you're meant to say it out loud, to somebody, not just write it. Now, that last part worries people. What if my words aren't the same as my friend's words?

They won't be. Take the triangular numbers. One person says: add one more each time than you added last time. Another says: add the next counting number. Those are different sentences. They are the same machine. Feed them both one, three, six, and they both hand back ten. So you're not being marked on phrasing. The rule is the answer, not the wording of it. Which brings us back to where we started. Look at how these rows begin.

Nine of the ten rows open with one. Only the even numbers don't. Three rows open one, two. The counting numbers, the powers of two, and the Virahanka numbers. The powers of two peel off immediately — one, two, four, while the others go one, two, three. And then the counting numbers and the Virahanka numbers part company at the fourth value. Four, and five. That's your answer from the cold open. Both were right. You just couldn't tell which one you were being shown.

It happens again with one, three. Three rows start that way: the odd numbers, the triangular numbers, and the powers of three. And all three split at the very next value. Five, six, nine. So no run of numbers, however long, ever pins down a sequence. Not two, not three, not seven. So: continuing a sequence is evidence for a rule. It is never a substitute for one. If somebody shows you numbers and asks what comes next, the honest first question is: which machine?

That's why your book asks for the rule in your own words, and not just for three more numbers. And that's the standard everything after this is held to. Here's my question for you. I'll give you a start, and you give me two different rules that both produce it. The start is one, two, four. One of them is on page three. The other one is yours to invent, and it has to be a real rule — something you can state, that keeps going for ever.

Put both in the comments. Next time, we take the first seven of these rows and draw them.

Where this fits

Taken from the notes each video was made from, not from the reading order — these are the ideas this one rests on and the ones that later rest on it.

Builds on

Comes up again in

The book

Open in a new tab