Skip to 0 minutes and 8 seconds MICHAEL ANDERSON: So let’s have a look now at the Fibonacci sequence.

Skip to 0 minutes and 11 seconds PAULA KELLY: So we have here the Fibonacci sequence. It’s a really special sequence. If we have a look, we have our first here is 1. Our second term is 1. Our third term is 2. So can you see where these numbers come from?

Skip to 0 minutes and 25 seconds MICHAEL ANDERSON: Not really, no. There doesn’t seem to be a pattern like some of the other sequences that we’ve seen where they go up by an equal amount or anything like that. You can’t multiply from one term to another, I don’t think. So yeah, it’s hard to see how this is formed.

Skip to 0 minutes and 38 seconds PAULA KELLY: So with this sequence, we’ve noticed perhaps that we had maybe 5 plus 8. What would that give you?

Skip to 0 minutes and 45 seconds MICHAEL ANDERSON: 13.

Skip to 0 minutes and 46 seconds PAULA KELLY: Very good. And then we had perhaps 2 plus 3. That would give you–

Skip to 0 minutes and 49 seconds MICHAEL ANDERSON: 5. So there seems to be some rule for generating it.

Skip to 0 minutes and 53 seconds PAULA KELLY: Absolutely. So to find the next term in the sequence, we add together the previous two terms.

Skip to 0 minutes and 58 seconds MICHAEL ANDERSON: Oh, OK.

Skip to 0 minutes and 59 seconds PAULA KELLY: So one common misconception is this double 1 to begin with. If we notice, though, we begin with a 1 because if we add a 1 and 0 together, we just end up with 1 there.

Skip to 1 minute and 10 seconds MICHAEL ANDERSON: Oh, OK. Yeah.

Skip to 1 minute and 11 seconds PAULA KELLY: So if we try and continue our sequence, this sequence will continue forever. So to find our next term in our sequence, if we put together 34 and 55, our next term in our sequence should be–

Skip to 1 minute and 24 seconds MICHAEL ANDERSON: 89.

Skip to 1 minute and 25 seconds PAULA KELLY: 89. OK. And again, our next term in our sequence–

Skip to 1 minute and 31 seconds MICHAEL ANDERSON: So 55. Add 89. That’s going to give us 144.

Skip to 1 minute and 36 seconds PAULA KELLY: Exactly. So we could continue forever and ever and so on. But we’re going to have a look a bit deeper in some more patterns with our Fibonacci style sequences too.

Skip to 1 minute and 46 seconds MICHAEL ANDERSON: OK. So let’s look in more detail about how to construct Fibonacci sequences.

Skip to 1 minute and 52 seconds PAULA KELLY: OK. So we can find what we call our term to term rule.

Skip to 2 minutes and 2 seconds So we’ll think about this. We know our first term in our sequence. We’re going to call that F1 because it’s the first one in our Fibonacci sequence. Can you remember our second term would be–

Skip to 2 minutes and 14 seconds MICHAEL ANDERSON: Well, it’s 1 again.

Skip to 2 minutes and 16 seconds PAULA KELLY: So 1 add 0 gives us 1. OK. And we want to find the term to term rule.

Skip to 2 minutes and 22 seconds MICHAEL ANDERSON: So that’s how to get from one term to the next term?

Skip to 2 minutes and 25 seconds PAULA KELLY: Absolutely. We’re going to call this the notation. We’re going to call it F. Fibonacci. n plus 1. how to find the next term in our sequence. So if we have with our sequence. To find the next term in our sequence, we add together our term and the term that came before it. So for this notation, we’ll have F of the one that came before it. We’ll call it n minus 1.

Skip to 2 minutes and 55 seconds MICHAEL ANDERSON: So if we were to try to find, for example, the sixth number in the Fibonacci sequence, n plus 1 would be 6, n would be 5, and n minus 1 would be 4.

Skip to 3 minutes and 6 seconds PAULA KELLY: Perfect.

Skip to 3 minutes and 7 seconds MICHAEL ANDERSON: So the Fibonacci sequence is a really famous sequence. Is it just one sequence?

Skip to 3 minutes and 12 seconds PAULA KELLY: There is just one Fibonacci sequence. We can generate, though, some Fibonacci style sequences. So if we use our term to term rule, we’ll see how that would work. So if we had, for example, our first term in our sequence, or F1, we could start with a 3. Our F2, our next one, could be a 2.

Skip to 3 minutes and 35 seconds MICHAEL ANDERSON: So we can pick any numbers, the first two numbers, and then generate a Fibonacci style sequence from there.

Skip to 3 minutes and 41 seconds PAULA KELLY: absolutely, yeah. So we’ve chosen any two numbers. We want to find the next term in our sequence. So for this, we’re going to have F3. Now, we’re going to use our term to term rule. If we’re finding out third term, we’re going to add together our second term

Skip to 3 minutes and 58 seconds MICHAEL ANDERSON: OK. So that’ll be F2.

Skip to 4 minutes and 1 second PAULA KELLY: And we’re going to add onto that our first term, the previous term.

Skip to 4 minutes and 6 seconds MICHAEL ANDERSON: OK. So F3 is equal to F2 plus F1.

Skip to 4 minutes and 10 seconds PAULA KELLY: Perfect. Our F2 is going to be– we know from here just 2. Our F1 is 3. So we can see our next term is going to be just 5.

Skip to 4 minutes and 23 seconds MICHAEL ANDERSON: OK. So this Fibonacci style sequence starts with 3, then goes to 2, and then 5. And the next term, I presume, would be F4. And that’s going to be 2 add 5, which is 7.

Skip to 4 minutes and 37 seconds PAULA KELLY: Fantastic.

Skip to 4 minutes and 37 seconds MICHAEL ANDERSON: And it’d keep on growing and growing and growing.

Skip to 4 minutes and 39 seconds PAULA KELLY: Forever, yeah.

# Fibonacci type sequences

The Fibonacci sequence is best described using a term to term rule and is how the sequence is described in school. There is a position to term rule but it is complicated and difficult to explain. It is beyond the scope of this course and is not usually taught in schools. If you are interested you can read the article on Binet’s Formula

It is worth noting that The Fibonacci sequence starts with the numbers 1, 1… We can create other ‘Fibonacci type’ sequence by using the same rule but starting with different numbers.

In this video Paula and Michael explain how to generate the Fibonacci sequence, how we write the term to term rule for the Fibonacci sequence and look at how we can create other ‘Fibonacci type’ sequences.

## Create

Play around with the starting numbers to create some Fibonacci type sequences. You may like to create a spreadsheet to help generate the sequences.

Can you find a sequence which just repeats the same number? Can you find any other interesting Fibonacci type sequences such as ones which loop around or repeat the same numbers?

Share your interesting sequences below.

## Problem worksheet

Now complete questions 2 and 3 from this week’s worksheet

## Teaching resources

- The magic of maths resource explores numbers in nature and discover that Fibonacci numbers and spirals frequently appear.
- The Honeybee family tree explains how to create the family tree for honeybees. The challenge begins with identification of male and female bees in a diagram. Further rows are then added and the information collected in a table. There is an extension to then find the number of bees in other generations without drawing the family tree. The challenge gives an introduction to the Fibonacci numbers.
- Leonardo of Pisa is an excellent extension resource which explores the ratio between terms of the Fibonacci sequence which tends to the Golden ratio.

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