MICHAEL ANDERSON: So let’s have a look now at the Fibonacci sequence.
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?
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.
PAULA KELLY: So with this sequence, we’ve noticed perhaps that we had maybe 5 plus 8. What would that give you?
PAULA KELLY: Very good. And then we had perhaps 2 plus 3. That would give you–
MICHAEL ANDERSON: 5. So there seems to be some rule for generating it.
PAULA KELLY: Absolutely. So to find the next term in the sequence, we add together the previous two terms.
MICHAEL ANDERSON: Oh, OK.
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.
MICHAEL ANDERSON: Oh, OK. Yeah.
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–
PAULA KELLY: 89. OK. And again, our next term in our sequence–
MICHAEL ANDERSON: So 55. Add 89. That’s going to give us 144.
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.
MICHAEL ANDERSON: OK. So let’s look in more detail about how to construct Fibonacci sequences.
PAULA KELLY: OK. So we can find what we call our term to term rule.
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–
MICHAEL ANDERSON: Well, it’s 1 again.
PAULA KELLY: So 1 add 0 gives us 1. OK. And we want to find the term to term rule.
MICHAEL ANDERSON: So that’s how to get from one term to the next term?
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.
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.
MICHAEL ANDERSON: So the Fibonacci sequence is a really famous sequence. Is it just one sequence?
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.
MICHAEL ANDERSON: So we can pick any numbers, the first two numbers, and then generate a Fibonacci style sequence from there.
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
MICHAEL ANDERSON: OK. So that’ll be F2.
PAULA KELLY: And we’re going to add onto that our first term, the previous term.
MICHAEL ANDERSON: OK. So F3 is equal to F2 plus F1.
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.
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.
MICHAEL ANDERSON: And it’d keep on growing and growing and growing.
PAULA KELLY: Forever, yeah.