PAULA KELLY: So let’s have a look at another number sequence. We have here some triangle numbers. And we’ll show you with some blocks to see how that looks.
PAULA KELLY: So what we have here is our fourth triangle number. Now, a triangle shape to our triangle number.
PAULA KELLY: You may notice it has a base of 4.
MICHAEL ANDERSON: Mm-hmm.
PAULA KELLY: So our fourth triangle number– we could do with knowing how many blocks are in this fourth.
MICHAEL ANDERSON: Yeah. So we can probably just count those.
MICHAEL ANDERSON: So we’ve got 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. So this is our fourth triangle number. It’s a height of 4 and a width of 4. And to make it, we need 10 blocks.
MICHAEL ANDERSON: So what’s the triangle number before that one?
PAULA KELLY: So as our fourth triangle number has a base of 4, height of 4, using our logic, our third triangle number has a base of 3, height of 3.
MICHAEL ANDERSON: Oh, so we’ve essentially just taken away this bottom row?
PAULA KELLY: Absolutely. Yeah.
MICHAEL ANDERSON: And if we look at the number of blocks we need to make this one, it’s just 1, 2, 3, 4, 5, 6.
PAULA KELLY: OK. So if we continue to it backwards, we have our fourth, our third, our second triangle number. Let’s keep our pattern going.
MICHAEL ANDERSON: Mm-hmm.
PAULA KELLY: It’s just going to have a base of 2, a height of 2.
PAULA KELLY: But this time, fewer blocks. We have just 3 blocks this time.
PAULA KELLY: OK. Finally, our first triangle number. As we’ve done all the way along, we’ve taken away 4, taken away 3, taken away 2. So for our first triangle number, just 1 block.
MICHAEL ANDERSON: Doesn’t look much like a triangle.
PAULA KELLY: [LAUGHS] It doesn’t do. But we can see from our pattern how it builds up into a triangle, and we’re seeing how we’ve gone backwards from the fourth triangle number right down to the first one.
MICHAEL ANDERSON: So usually, with these sequences, we normally work up. So we’d start with 1, and then we’d add something, and add something, and add something again. So how does this one grow, working upwards?
PAULA KELLY: So if we notice, we start from our first triangle number. We have just 1. We notice with our numbers, we add on 2. But showing it with our block makes it even clearer. We can see we’ve added an extra row. That’s an extra 2 blocks.
MICHAEL ANDERSON: OK. So to get from 1 to 3, we add 2.
PAULA KELLY: OK-doke. OK. So looking at our pattern again, we’ve added on an extra row at the bottom. We’ve added on an extra 3 blocks.
MICHAEL ANDERSON: OK. So we’re adding 3 on this time. So it’s not an arithmetic sequence, because this time we’re adding on 3 instead of 2. The differences are different.
PAULA KELLY: Exactly. So for an arithmetic sequence, we have to go up by the same amount each time. So for triangle numbers, we can see now, it’s not necessarily an arithmetic sequence.
MICHAEL ANDERSON: OK. And to get from 6 to 10, we added this last row in, so we’re going to add 4. So it seems to me that the amount we’re adding on each time is increasing by 1 each time. So to get the fifth triangle number, we probably add 5. And then add 6, and then add 7, and add 8. And keep on going to get the next term, and the next term, and the next turn.
PAULA KELLY: Yeah. Forever and ever and ever. OK. Yes.
MICHAEL ANDERSON: OK. So now we know how to get from one term to the next term, let’s see if there’s a general rule for the triangle numbers.