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Fibonacci type sequences

The Fibonacci sequence is best described using a term to term rule and is how the sequence is described in school.
7.6
MICHAEL ANDERSON: So let’s have a look now at the Fibonacci sequence.
11.1
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?
24.5
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.
37.7
PAULA KELLY: So with this sequence, we’ve noticed perhaps that we had maybe 5 plus 8. What would that give you?
44.9
MICHAEL ANDERSON: 13.
45.5
PAULA KELLY: Very good. And then we had perhaps 2 plus 3. That would give you–
49.1
MICHAEL ANDERSON: 5. So there seems to be some rule for generating it.
52.6
PAULA KELLY: Absolutely. So to find the next term in the sequence, we add together the previous two terms.
57.5
MICHAEL ANDERSON: Oh, OK.
58.7
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.
69.5
MICHAEL ANDERSON: Oh, OK. Yeah.
71.2
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–
84.2
MICHAEL ANDERSON: 89.
84.8
PAULA KELLY: 89. OK. And again, our next term in our sequence–
90.6
MICHAEL ANDERSON: So 55. Add 89. That’s going to give us 144.
96.2
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.
105.8
MICHAEL ANDERSON: OK. So let’s look in more detail about how to construct Fibonacci sequences.
111.8
PAULA KELLY: OK. So we can find what we call our term to term rule.
122.3
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–
134.4
MICHAEL ANDERSON: Well, it’s 1 again.
135.9
PAULA KELLY: So 1 add 0 gives us 1. OK. And we want to find the term to term rule.
141.7
MICHAEL ANDERSON: So that’s how to get from one term to the next term?
144.6
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.
174.6
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.
185.7
PAULA KELLY: Perfect.
186.9
MICHAEL ANDERSON: So the Fibonacci sequence is a really famous sequence. Is it just one sequence?
192.4
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.
215.4
MICHAEL ANDERSON: So we can pick any numbers, the first two numbers, and then generate a Fibonacci style sequence from there.
220.7
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
238.3
MICHAEL ANDERSON: OK. So that’ll be F2.
240.5
PAULA KELLY: And we’re going to add onto that our first term, the previous term.
245.9
MICHAEL ANDERSON: OK. So F3 is equal to F2 plus F1.
249.5
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.
263
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.
276.5
PAULA KELLY: Fantastic.
277.4
MICHAEL ANDERSON: And it’d keep on growing and growing and growing.
279.3
PAULA KELLY: Forever, yeah.
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 that 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 the 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 that explores the ratio between terms of the Fibonacci sequence which tends to the Golden ratio.
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Maths Subject Knowledge: Understanding Numbers

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