Lesson video

Thank you for joining me for today's lesson.

My name is Ms. Davies, and I'm gonna be guiding you as you explore some of these new and exciting sequences that we're looking at today.

Make sure that you've got everything you need before you start watching this video.

It's always a good idea to have a pen and paper so that you can jot things down and explore things in your own time.

Let's get started, then.

Welcome to this lesson on checking and securing understanding of special number sequences.

By the end of this lesson, you'll be able to identify features of these special number sequences.

With that in mind then, we need to be confident as to what a square, a cube, and a triangular number are.

We'll revisit these in the lesson, but if you want to pause and remind yourself, please do that now.

And we're going back to some of our sequences work.

So a sequence is an ordered list of items, usually formed according to the rule.

We're gonna mention the nth term rules today.

The nth term of a sequence is the position of a number in a sequence, where n stands for the term number.

We might also mention some term-to-term rules.

So a term-to-term rule describes how to calculate the next term from the previous term.

So we're gonna start by looking at square, triangular, and cube number sequences.

So Aisha is investigating a sequence with first four terms, 3, 7, 11, 15.

She says, "These could be the first four terms in an arithmetic sequence." What does she mean by this? What do you think? Well, we can see that between 3 and 7, we're adding 4, then we're adding 4 again, then we're adding 4 again.

There is a common difference of four between successive terms. Now arithmetic sequences have a common difference between successive terms. You might have heard them called linear sequences as well.

Now Aisha has no way of knowing whether this pattern will continue, which is why she said that these could be the first four terms in an arithmetic sequence.

So now she's looking at a new sequence, which starts 3, 9, 27, 81.

These cannot be the first four terms in an arithmetic sequence.

How can she tell? Right, we need to check for a common difference, or from 3 to 9, we're adding 6, but then we're adding 18 and then 54.

So they do not have a common difference.

This is not an arithmetic sequence.

Lucas says that these could be the first four terms in a geometric sequence.

Is Lucas correct? Can you remember what a geometric sequence is? Does that apply to these four values? Well done if you remembered that a geometric sequence has a common multiplier between terms, 3 times 3 is 9 times 3 is 27 times 3 is 81.

So we have got a common multiplier, which is called a common ratio between successive terms. So we've just revisited two types of sequence, arithmetic and geometric, but there are other types of sequences as well, and we can recognise those from some of their key features.

So Izzy has written the first four terms of a different sequence.

She's written 1, 4, 9, 16.

What do you notice about these numbers? Pause the video.

What can you spot? Well, you might have noticed that they are the first four square numbers.

It was one of our key words, so that might have given you a clue.

So those are our term numbers.

Term one, term two, term three, term four.

Then I've written the term underneath, and sometimes putting a sequence in a table can help you see what's happening.

If her sequence is all the square numbers in ascending order, what would the 10th term be? Right, well, the 10th term would be the 10th square number, which is 10 squared, or 100.

What would be the rule then for finding the nth square number? So to find any square number, what do we need to do? Can you write it as a rule? Well, to find any term in the sequence, we square the term number.

So the nth term rule of all square numbers will be n squared.

Izzy says, "The sequence of square numbers is not arithmetic or geometric." She's absolutely correct, there is no common difference by adding three then five.

There's not a common ratio.

We're multiplying by four, but 4 times 4 is not 9, so it's not arithmetic or geometric.

But can you see a pattern with how the sequence is growing? Can you describe a term-to-term rule? There's the first six square numbers to help you.

Well done if you spotted that we're adding 3, then 5, and 7, then 9, then 11.

So there's definitely a pattern with the difference between the terms. The difference is increasing by two each time.

Sequence adds 3, then 5, then 7, then 9.

Now we can call those the first difference 'cause they're the difference between the terms. Then we can refer to the difference between the differences as the second difference.

What we can say then is this sequence has a common second difference of two, and that language is just gonna help us describe these sequences.

Right, if we construct the square numbers with dots, we can see this pattern.

So those are the first four square numbers, and we can see that we're adding 3, then 5, then 7, and we know that's then going to continue.

Right, how could we then work out the next term in the sequence after 36? What do you think? Well, if our differences are increasing by two, next time we're gonna need to add on 13, which is 49.

The other way we can do it is we could find the seventh square number.

We've already said that the rule is n squared, so if we do seven squared, we get 49 as well.

Right, quick check, then.

Here are the 19th, 20th, and 21st terms in the sequence of all square numbers.

Using whichever method you like, what is the 22nd term? So we could look at the differences.

We're adding 39, then 41, so the next time we're gonna add 43.

That gives us 484.

Or you could have done 22 squared if you preferred.

Right, now we're gonna look at triangular numbers.

So these are numbers that can be represented by a pattern of dots arranged into an equilateral triangle.

So there's the first term, the second term, the third term, and you can see that we have three dots on the height, three dots on the base.

What are the next two triangular numbers? Well done if you said 10 and then 15.

If we look at some of the patterns, you can see that each time we're adding an extra row that has one more in than the previous row.

Can you describe, then, how this pattern is growing? What do you notice? You can see we're adding 2, then 3, then 4, then 5, so the difference is increasing by one each time.

If we return to our language from before, we have a common second difference of one.

How could we work out the next term in the sequence? What do you think? Well, our differences are increasing by one, so if we add seven next time, we get 28.

Now the nth term rule is not as easy to work out.

For now, we can just use the term-to-term rule to generate the terms in order.

Let's remind ourselves about cube numbers.

The cube number is the product of three repeated integers, so 2 times 2 times 2 is 8, so 8 is a cube number.

We can build cube numbers out of cubes, and so we've got a cube there with dimensions 2, 2, and 2.

If we look at 27, 27 is a cube number, and we can arrange 27 blocks into a cube with dimensions 3 by 3 by 3, and that gives us 27.

Lucas says, "I wonder if the sequence of all cube numbers have a common second difference too." Let's have a look together.

Well, there are our first six cube numbers, and those are the first differences.

We're adding 7, the 19, and 37, and 61, then 91.

Ooh, but that doesn't follow a linear pattern.

The second differences this time is 12, then 18, then 24, then 30.

They're not all the same, so they do not have a common second difference.

I wonder if you are as quick as Izzy, and has noticed that they do have a common third difference.

See if you can use that pattern then to work out the next cube number.

This is gonna be a little bit trickier, so well done if you get this.

Right, well, if we start at the bottom, so we're gonna need to add six to our second difference to get a second difference of 36.

So 91 add 36 is 127, so we need to add on 127.

That was tough.

Lucas is right.

It would've been easier to use the nth term rule.

The nth term rule for all the cube numbers is just n cubed.

We cube the term number, we get the cube number for that term.

So the seventh term would be 7 times 7 times 7, so 49 times 7, which is 343.

You can see with the cube numbers, they get really big really fast, which is why it's quite hard to work out the next term.

Even using that nth term rule can be quite tricky if we find it difficult to cube our numbers.

Using a calculator will definitely help us with this one.

Right, here are the 12th, 13th, and 14th terms in the sequence of all triangular numbers.

What is the 15th term? So let's look at our differences.

We're adding 13 then 14.

Triangular numbers have a common second difference of one.

So if we add 15, we get 120.

For now, we don't know the nth term rule of the triangular numbers, so we can't use that to help us.

You may have noticed that for triangular numbers, the difference between a term and the previous term is the same as the term number.

So to get the 15th term, we can add 15 onto the 14th term, so that's gonna help us get the next triangular number.

Right, time for a practise.

I'd like you to fill in the missing numbers in these special number sequences.

So you've got your triangular, your square, and your cube numbers.

And then you've got a Venn diagram.

I'd like you to find one number less than 100 to go in each section of the Venn diagram.

So you've got all the different sections within the three circles and the overlapping parts between them.

And can you find a number that will go in the region outside the circles within that rectangle? One of the regions is impossible.

See if you can find which one.

Time to put that into practise, then.

Have a go at these five questions.

Question four, Aisha is drawing staircase patterns on square paper.

You can see her first, second, third, and fourth pattern.

If she continued to build these in the same way, how many squares will be in the eighth pattern? How do you know? Give this one a go.

Right, Lucas is investigating a different pattern.

This is the mystic rose pattern.

So if you draw a circle with six points equally spaced around the circle and then join each dot to all the other dots with a straight line, you get this pattern.

I'd like you to have a go at drawing the mystic rose pattern, but for 2, 3, 4, and 5 dots.

Then tell me how many lines are in Lucas's pattern for six dots.

And can you work out how many lines will be in the pattern for 12 dots? You should be able to do this without drawing it.

How do you know? Let's have a look then.

So pause the video and check your triangular, square, and cube numbers.

Let's have a look at this Venn diagram, then.

So one is the only number that's gonna go in the very middle.

It is a square, cube, and triangular number.

The only other number that is both square and cube under 100 is 64.

Obviously, one is as well, but we can't put one in that section because it's also a triangular number.

So that needed to go in the middle section.

The only other number that's square and triangular under 100 is 36.

Now there isn't another number, which is both a cube number and a triangular number.

It's not too difficult to see that for 100 because you can see your cube numbers up to 100.

In fact, it's not possible even beyond 100.

That might be something you want to investigate.

Then you've got lots of choices for square numbers that aren't triangular or cube.

I've given you some suggestions, 9, 81, 49.

Again, cube numbers, you can go with 8 and 27.

And triangular numbers, there's loads of them under 100.

You could have gone with 3, 6, 10, or so on.

The outside section, make sure that you get a number that is not a triangular number.

There's quite a few triangular numbers under 100, so make sure the ones that you pick are not triangular.

If you go with two, you know you're safe 'cause you can see that's not in your triangular, or square, or cube number sequences.

Question three, the easiest way to the triangular numbers is to keep generating them until you get the 10th triangular number.

That's if we're not able to find the nth term rule, which we can't at the moment.

So the 10th triangular number is 55.

It's 10 add 9 add 8 add 7 add 6 add 5 add 4 add 3 add 3 add 1.

Right, the 10th term in the sequence of all square numbers.

Well, if we do 10 squared, that's 100.

10th term in the sequence of all cube numbers, 10 cubed is 1000.

If the 23rd triangular number is 276 and the 24th is 300, what's the 25th? We can just add 25 onto 300, which gives us 325.

With the square numbers, you could do 25 squared is 625.

You might also notice, and this is quite an interesting fact, that consecutive triangular numbers add to a square number.

So actually, the 25th square number is the 24th triangular number, add the 25th triangular number.

Well, we've just calculated those above.

So you could just do 300 and 325.

Again, if you want to pause and investigate that further, you could do so now.

And finally, you may have noticed that this is the sequence of square numbers.

Each row has two more squares than the previous row, so the pattern has a second difference of two.

The eighth pattern will be the eighth square number, which is 64, so it'll have 64 squares.

I hope you enjoyed playing around with this pattern.

You can do it with more and more dots.

You've just gotta make sure you get those dots equally spaced around the outside of your circle.

So how many lines are in Lucas's pattern for six dots? There should be 15.

Now I wonder if you noticed the pattern.

We had 1, then 3, then 6, then 10, then 15.

So these are the triangular numbers.

Gotta be a bit careful here, 'cause we started with two dots.

So 12 dots is actually the 11th term in the sequence, 'cause two dots was the first term.

So 12 dots is the 11th term, and the 11th triangular number is 66.

Hope you had a bit of fun playing around with that sequence.

You do find the triangular numbers in lots of different places in mathematics.

Right, we're gonna play around then with some disguised sequences.

We've seen how the triangular numbers in ascending order and the square numbers in ascending order form sequences with common second differences.

There's the triangular numbers and the square numbers.

But there are other sequences that have this feature as well.

We call them quadratic sequences.

It's quite useful to know that word 'cause we can apply it then to this type of sequence.

So a sequence with a common second difference is a quadratic sequence.

Let's have a look at this one.

3, 6, 11, 18, 27.

Pause the video.

Can you see any patterns with this sequence? Right, well, just like another type of number we've seen, it adds 3, then 5, then 7, then 9.

So you might have said something like, "It is a quadratic sequence," if you just learned that new word.

It has a common second difference of two.

And you might have noticed this, each term is two more than the square numbers.

The first square number is one, but our first term is three.

The second square number is four, but our first number is six.

So there's our square numbers, 1, 4, 9, 16, 25, and there's our sequence 3, 6, 11, 18, 27.

For each term, we've added two.

So if we call that first sequence n squared, we can call this sequence n squared plus 2.

Izzy says, "The square numbers have a common second difference of two, and so does this sequence." It's that pattern we're gonna explore today.

Lucas says, "What would happen if we subtracted one from the square numbers?" Well, let's have a look.

We get 0, 3, 8, 15, 24.

Let's check the common second difference for adding 3 then 5 and 7 then 9, so our differences are increasing by two.

What about if we doubled the square numbers? If we double our square numbers, we get 2, 8, 18, 32, 50.

Pause the video.

What patterns can you notice in this sequence? Right, we're adding 6, then 10, then 14, then 18, so we do have a common second difference, but this time it's four rather than two.

So Lucas has spotted something.

Adding a constant to the square numbers kept the second difference the same, but multiplying the square numbers did not.

Izzy says, "I wonder what would happen if we added 10 to each of the triangular numbers." Pause the video, make a prediction about what you think that pattern will look like.

Well, there's our triangular numbers.

Let's add 10 to each.

We're adding 2, then 3, then 4, then 5, which is a common second difference of one still.

So the new sequence is 10 more than the triangular numbers, and it still has a common second difference of one.

In fact, it has the exact same first differences for adding 2, then 3, then 4, and so on.

Right, quick check.

I'd like you to match the sequences to the descriptions.

I've given you the first five triangular numbers to help you.

Off you go.

So A is the triangular numbers multiplied by 10, B is the triangular numbers add nine, C is the triangular numbers add one, and D is the triangular numbers multiplied by two.

Right, what is the second difference, then, for each of these sequences? Give it a go.

Right, we're adding 20, then 30, then 40, then 50, which is a common second difference of 10.

Notice that multiplying the triangular numbers by 10 meant the second difference was multiplied by 10.

For the second one, we've got a common second difference of one.

And the same for C.

For these two, we've added a constant to every term in the triangular numbers, so it hasn't changed our common second difference.

And for D, we've got a common second difference of two.

Multiplying each value by two meant the second difference was multiplied by two.

You don't have to memorise those patterns at the moment, but starting to explore those will help you when you do more work with these sequences in the future.

Successive cube numbers do not have a common second difference, so they do not form a quadratic sequence.

We can still use the general features to generate other sequences.

Pause the video.

Can you spot how this sequence is related to the cube numbers? Right, well, writing our cube numbers out is gonna help us.

Then, if we can compare them to our sequence, they are the cube numbers add four.

They do not have a common second difference.

They are not a quadratic sequence.

Time for a practise.

I'd like you to match the first four terms of the sequences with their descriptions.

Give it a go.

For question two, each of these sequences are related to either the triangular, square, or cube numbers, so you might wanna start by making sure you've written out your triangular, square, and cube numbers.

Now I want you to suggest what the next term in the sequence would be, but you need to explain your answer.

Give those a go.

Question three, the table shows the 20th, the 21st, and the 22nd term in a sequence.

Aisha says, "The sequence could be the square numbers subtract 50." Can you explain why she must be incorrect? Right, let's have a look at these answers then.

A is the triangular numbers subtract one, B, the square numbers add four, C, the square numbers multiplied by three, D, the triangular numbers doubled, then three is added.

That was a tricky one.

E is the cube numbers add two, and F is 10 less than the cube numbers.

For question two, you might have come up with slightly different rules.

We're looking for rules that relate these to the triangular, square, or cube numbers, though.

So for A, I think these are the square numbers add seven, so the next term would be 32.

For B, I think these might be the triangular numbers multiplied by eight, in which case the next term would be 120.

C could be the cube numbers multiplied by five, so the next term would be 625.

D, it could be the square numbers divided by two, so the next one's 12.

5.

This one was tricky.

Well done if you got this.

I think this could be the triangular numbers subtract to half.

You might have found that turning them into decimals helped you with this.

The next one, then, is 29 over 2.

And F, I think the square numbers multiplied by four, then subtract one.

That was really tricky to spot, so well done if you saw that.

The next one would be 99.

If you got a different pattern related to the square, triangular, or cube numbers, that's absolutely fine as long as you explained your reasoning.

For three, there's a couple of ways could have gone about this.

You could have looked at the differences.

We're adding 40, then 50, so we've got a second difference of 10.

Well, the square numbers have a second difference of two, and just subtracting 50 from the square numbers won't change our second difference, so they can't be the square numbers subtract 50.

Alternatively, you could have tried it.

You could have gone with 21 squared subtract 50, 22 squared subtract 50, and shown that that doesn't work for the 21st and the 22nd term.

So let's recap what we've looked at today.

We know that there are sequences that follow rules, which are arithmetic or geometric, and we had a recap as to what those two terms mean.

But we know there are sequences that follow rules which are not arithmetic or geometric, and we explored those in more detail.

The square and triangular numbers are examples of sequences with a common second difference.

The cube numbers form a sequence with a common third difference.

And we can use this feature, especially the fact that the square numbers have a common second difference, to find related sequences.

So we looked at sequences where we were adding numbers to each of the terms and we were multiplying each of the terms by the same number, and seeing that it still has this feature of a common second difference.

If you want to know that new word was quadratic sequences, a quadratic sequence has a common second difference.

Thank you for joining us today.

I hope you enjoyed playing around with some of those patterns, and maybe you'll keep your eye out for triangular numbers when they appear in other areas of mathematics.