Lesson video

Hi, I'm Mr. Chan, and in this lesson we're going to look at triangular and Fibonacci style sequences.

Let's begin by looking at the triangular sequence.

I've got some counters here to show the first three triangular numbers.

Look at how many counters we have.

We start with one, then three, then six, and you can notice I've set these counters up in a particular pattern.

We can put a triangle around these counters, and that's one of the reasons why this sequence is called the triangular numbers, or the triangular sequence.

What are the first three triangular numbers? So we have one, then three, then six.

Do you see how this sequence is building up? What would the fourth triangular number be? What the fourth triangular number would be is 10.

And what's happening each time with the triangular numbers is we start with one and we can add two to get three, then we add three to get six, then we add four to get 10.

So the sequence is building up by adding one more each time.

So we start with adding two, then we add three, then we add four, so the next one would be add five.

So do you know what the next triangular number would be? 15, that's correct.

Here's a question for you to try.

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Here are the answers.

We looked at generating the triangular numbers in the example.

So you'll see here the triangular numbers are generated by adding one more each time.

Another way of describing that more mathematically is they're formed by adding consecutive integers starting with one.

Here's another question for you to try.

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Here are the answers.

This is a lovely question to demonstrate why triangular numbers are quite important in maths.

They appear in lots of different areas in real life, in particular in combination counting.

So when we look at how many combinations of jelly beans people can get in this question, we see the numbers one, then three, then six, then 10.

And you can spot there that the sequence that are generated are the triangular numbers.

Let's now look at the Fibonacci sequence.

I've got some cubes to show the first four terms of a Fibonacci type sequence.

So we'd begin with one.

We have another one, then two, then three.

So with this sequence, how do you think this sequence is growing? We've got one, another one, two, and then three.

So let me explain.

What's happening with this sequence is each term is the sum of the previous two terms. So we take the first two terms, one and one, we add those together we will get two.

Then we will take the one and the two, add those together, we get three.

So we're summing the previous two terms to create the next term.

So what do you think the next term is going to be? Well, let's have a look.

We're going to take the previous two terms, which would be the two and the three, add those together, let's see what we get.

So I take the two, take the three, add those together, and we get five.

Well done if you got that right.

Here's an example of a Fibonacci style sequence.

We've got to work out the next three terms in this sequence.

So we're told it's a Fibonacci style sequence, so what happens with this is the next term is the sum of the previous two terms. So we've got the starting points, two, two, four, and six.

So let's just check it's working out.

Two add two, we would get four.

Two add four, we would get six.

So the next term I would add the four and the six together, so four add six, we would get 10.

Next term would be the sum of the previous two terms, so this would be the six add 10, 16.

Next term, the sum of the previous two terms, so we've got 10 and 16.

10 add 16, 26.

That's how Fibonacci style sequences grow.

Here are some questions for you to try.

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Resume the video once you're finished.

Here are the answers for question three.

Lots of Fibonacci style sequences there, so they're not just limited to one type.

Now, let's look at question B, for example, hopefully you've got that correct.

To generate that sequence, it has told you it's a Fibonacci style sequence, so the two and the five we would add to get the seven.

Then, to get the 12, we add the five and the seven.

The next term would be the seven add 12 to get 19.

Then 12 add 19 to get 31.

Then 19 add 31 to get 50.

So you can see the Fibonacci style sequences are adding the previous two terms. Now, Fibonacci style sequences don't always have to be ascending either.

So in part C there's an examples there where the Fibonacci style sequence involves some negative numbers, so hopefully you've got that correct, but I wonder if you could continue that Fibonacci sequence and see where that ends up.

Here's some more questions for you to try.

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Here are the answers.

Now this question asks you to find some missing numbers in a Fibonacci style sequence.

So if we look at part a as an example, hopefully you got that correct, but with part a, I would have started maybe with looking at the 15 in the middle.

Now, I know that I have to add the previous two terms to get the 15, and I do only have one of the terms, which is the six.

So in order to get 15 with six, what do I need to add? I would have to add nine, which is why the missing number there in between the six and the 15 is nine, because then I know six add nine gets me the 15.

So this type of question involves working backwards a little bit.

That's all for this lesson.

Thanks for watching.