Lesson video

Hi there.

I am really glad that you've decided to make the decision to learn with us today.

My name is Ms. Davies and I'm going to help you as you work your way through these lessons.

If there's anything you're not sure of, take your time to pause the video, have a little bit of a think, and then we'll go through answers together.

Let's get started then.

Welcome to this lesson on recognising special number sequences.

We're going to be exploring lots of different types of number today and having a look at the sequences they form.

By the end of the lesson, you'll be able to recognise a special number sequence.

We're going to have a look at some interesting key features of some of these sequences.

If you're not sure about an arithmetic or a geometric sequence, please pause the video and have a read through.

I want to draw your attention to our third key term, which is a triangular number.

A triangular number is a number that can be represented by a pattern of dots arranged into an equilateral triangle.

The term number is the number of dots in the side of the triangle.

So for example, 15 is the fifth term.

It's got five dots along the base of the triangle, five on the left hand side of the triangle, and five on that diagonal side of the triangle.

The important thing is that each row has one more dot than the row above it.

So let's start by having a look at arithmetic and geometric sequences.

An arithmetic, or a linear sequence, has a common difference between successive terms. So I'm going to give you the first five terms of some sequences.

What could we do to see if they are arithmetic? What do you think? Perfect.

Hopefully you said something like, work out the difference between successive terms and check it's the same all the way through our sequence.

So we want to know that the difference between 81 and 138, it's the same as the difference between 138 and 195, and so on.

Once we've checked our terms, that's enough to say that they are at least following a linear rule for the first five terms. Let's work through these together.

So for a, if we do 138 subtract 81, that's 57.

We need to check the next two terms. Also 57.

Aisha says it must be arithmetic then.

What do you think to her statement? So Laura says, "Actually, we need to check the other terms." So let's see what the other terms look like.

Okay, so now we've checked all five terms. So far, it looks like an arithmetic sequence.

That's not to say that something won't change, but they are definitely the first five terms of a sequence which could be arithmetic.

Let's have a look at this second one.

So 13 subtract negative 24 is 11.

So negative 24 add 11 gets you to 30.

We'll do the same for the next two terms, and the next two terms, and finally the last two.

Right, we can see now then that this one is not arithmetic.

It starts off looking like it's adding 11 each time, but then we've got an add 15 in the middle.

Okay, pause the video, have a think about the last two.

What would you do to check those? And then come back to the answers.

Okay, so we've got 19.

4 subtract 21.

7, and that's negative 2.

3.

So that difference there is we're subtracting 2.

3.

Laura says, "I find it easier to keep subtracting 2.

3 and see if I get the next term." That's a perfectly fine way of doing this.

So we could do 17.

1 subtract 19.

4, or we could just subtract 2.

3 from 19.

4 and see if we get 17.

1.

We do get 17.

1.

Subtracting 2.

3 again gets us 14.

8, and subtracting 2.

3 again gets us 12.

5.

And finally, the last one.

This might be a little bit easier if we convert them over to a common denominator.

So I've chosen a common denominator of six, and then we're subtracting 1/6, subtracting 2/6, subtracting 1/6, and subtracting 2/6.

So, no, that one is not an arithmetic sequence.

So Lucas has got a sequence.

"I have started a sequence.

Eight, 12, 18, 27." Is Lucas' sequence arithmetic? Try it out.

You should have seen then that, no, the difference between the first two terms is four, but then that does not stay constant.

What other types of sequences could it be? So Aisha says, "Maybe it's geometric." What could we do to check whether Lucas' sequence is geometric? A bit of a test to make sure you are happy with these keywords.

Okay, so what we need to do is divide successive terms and see if there is a common ratio.

So we've got eight, 12, 18, 27.

Can we multiply by the same value to get the next term? So let's try doing 12 divided by eight, and I'm going to write that as a fraction and simplify it.

So that gives us three over two.

Let's try 18 divided by 12, three over two again.

27 over 18, they're both divided by nine.

27 divided by nine is three.

18 divided by nine is two.

So we can simplify that to three over two.

Lovely, so far then, these could be the first four terms of a geometric sequence.

We've got a common multiplier, sometimes referred to as a common ratio, of three over two.

Time for a check.

So which of those could be the first four terms in a geometric sequence? What do you think? Welcome back.

Let's have a look at our answers.

So we're multiplying by four, then by four.

So we need to check if 48 times four is 194.

It's not.

It's 192.

We can tell that because eight times four is 32, therefore, our number will end at a two, not a four.

Multiplying by five and by five and by five again.

So, so far, yeah, we've got a common ratio of five.

We're multiplying by five over two, five over two, and five over two.

So, so far, those form a geometric sequence.

Then here, we're multiplying by negative two, then by two, then by two.

So not a common multiplier, because we're multiplying by negative two and then by two.

So now we're going to put that together with some of our algebra skills and have a look at some sequences that have algebraic expressions as terms. So we've got a sequence here which starts a plus three, that's the first term, a plus seven is the second term, a plus 11 is the third term, and a plus 15 is the fourth term.

That could be an arithmetic sequence.

Let's have a look.

It has a common difference of add four.

A plus three add four gives you a plus seven.

Add four gives you a plus 11.

So although they're algebraic expressions, they still form an arithmetic sequence.

Have a look at this one.

A plus b, a plus three b, a plus five b.

What do you think the next term would be if it's an arithmetic sequence? So, first, we need to find our common difference.

Hopefully, you got that as plus two b.

Then if we add two b on, we've got a plus seven b.

You are really showing now that you've got this understanding of how arithmetic sequences are formed.

So Laura says, I think a, two a, three a, four a is a geometric sequence, because it uses multiplication to get the next term.

Is Laura correct? What do you think? Right, this one does fool quite a lot of people, because yes, if you multiply a by two, you do get two a.

But then what happens when you multiply two a by two? Because remember, we need this common multiplier.

So if we multiply a by two, we get two a, but two a multiplied by two should give us four a, not three a.

In fact, this is actually an arithmetic sequence.

What we're doing is we're adding a each time.

So if it was a geometric sequence like Laura wanted it to be, if it starts a, two a, the next term is going to be four a.

Then if we multiply that by two, we get eight a.

Then if we multiply that by two, we get 16 a.

So we've got a two a, four a, eight a, 16 a.

They could be the first five terms of a geometric sequence.

Perfect.

Have a look at these three then.

Which of these could be geometrics? We want to think about your algebra skills.

Think about if there's something we are multiplying by each time.

Take your time and then we'll check our answers together.

So, the first one, we're multiplying by five, then we're multiplying by two, then we're multiplying by five.

So that's not geometric.

This second one, a times a is a squared.

If we multiply by a, again, we get a cubed, and if we multiply by a again, we get a to the power of four.

So well done if you spotted that did have a common ratio of a.

This one, so to get from a to ab, you multiply by b.

Then to get to two ab, we'll multiply 'em by two.

Then we're multiply 'em by two again.

So not geometric that last one.

Well done.

Time for you to bring all of that together.

So for this task, you've got some sequences down the side of the page.

I would like you to sort them into the correct column in the table.

So they may be arithmetic, they may be geometric.

For now, if they're not one of those two, can you put them in the other column? You might want to think about any patterns you can spot, why it is that you didn't sort it into the other two columns.

Off you go and come back to the next bit.

Well done.

Looking like we're real experts now on these arithmetic and geometric sequences.

We're going to do the same this time.

A lot of these sequences contain fractional terms. Give yourself space to do your working out.

You might want to make sure you are converting over a common denominator to see if you can spot those common differences or common multipliers.

Off you go and then we'll look at the next bit together.

Well done, so now you've brought your fraction skills in with your sequences skills.

We're going to do the same with our algebra skills.

So a sequence starts two a, six a.

If the sequence was arithmetic, what would the next three terms be? Then do the same for geometric.

Extra special challenge at the end, continue the sequence in another way, and then describe the rule you have used.

See if you can be as creative as you can.

You might want to think of a couple of different ways you might want to test with a partner if you've got a learning partner, and see if you can work out each other's sequences.

Off you go and then come back for the answers.

Well done, so in the arithmetic section, you should have one, two, three, four, five, negative one, five, 11, 17, 23, 36, 12, negative 12, negative 36.

For geometric, three, nine, 27, 81.

We've got a multiplier of three.

0.

5, one, two, four.

We've got a multiplier of two.

Negative two, four, negative eight, 16.

We've got a multiplier of negative two.

13.

5, nine, six, four.

We've got a multiplier of 2/3.

Lovely, so the remaining four will be in that neither column.

If you came up with some ideas of how the sequences are formed, keep those in your head because they might come up again in the next part of the lesson.

For our fractional sequences, three over two, one, half, and zero is arithmetic.

We're subtracting a half.

Negative 4/5, negative 2/3, negative 18/15, and negative 2/5 is arithmetic.

We are adding 2/15 each time.

B is geometric.

We're doubling our sequence each time.

C is geometric.

We're multiplying our sequence by 10 each time, and e is geometric.

We've got a common multiplier of negative a third.

The other two are in our neither column for now.

And finally, if it was an arithmetic sequence, we'll be adding four a.

So we have 10 a, 14 a, 18 a.

If the sequence was geometric, we're multiplying by three.

So 18 a, 54 a, and 162 a.

Absolutely fine if you wanted to use a long multiplication method to help you with those larger multiplication.

There are so many possibilities of things you could have come up with for the last one.

You could have added the two previous terms to get the next term.

You could have started by adding four a and then added five a and then added six a.

All sorts of things you could have done with those two.

Let's have a look now at some other sequences.

We're going to explore some more interesting sequences that you may not have come across before.

So there are other types of sequences we can recognise from their key features.

So here we've got three, six, 11, 18, 27.

We can see it's not geometric.

It starts by multiplying by two, but that doesn't continue.

We can see it's not arithmetic.

It adds three and five and seven, then nine.

There does seem to be a pattern though.

Could you put into words the term-to-term rule for this sequence? You may have said something like the sequence adds three, then five and seven, then nine.

The sequence is adding the odd numbers, or there's a linear pattern in the differences.

The differences seem to increase by two each time.

What we can say is that there's a common second difference of two.

The differences are not the same, but the difference in the differences are.

So a common second difference of two.

I wonder if, like Izzy, you are thinking, "I've seen this before." If you've looked at the square numbers, the square numbers do something similar.

So if we look at one, four, nine, 16, 25, there are square numbers, and they increase by three, then by five, then by seven, then by nine.

So this sequence of all square numbers has an nth term rule of n squared.

The square in the term number gives the term.

Have a look at this sequence.

Three, six, 11, 18, 27, the one we started with.

We can see that it's actually our square numbers, but add two.

You might have done something similar before with linear sequences where you find the times table that it's related to and then see how much has been added or subtracted.

So if we're calling the top sequence n squared, this new sequence has nth term rule n squared plus two.

Lucas says, "What would happen if we subtracted two from the square numbers?" Let's have a look then.

So there's our square numbers.

If we subtract two, we've got the new sequence n squared subtract two.

Let's see if it has a common second difference.

So we've got add three add five add seven add nine, and it's got a common second difference of two again.

Let's do the same with doubling our square numbers.

So we've got two, eight, 18, 32, 50, and if the top sequence is called n squared, doubling it could be called two n squared.

This time though, we're adding six, then 10, then 14, then 18.

Okay, so this time we've got a common second difference of plus four.

The more you investigate these sequences, the more patterns and rules like this you're going to discover.

Izzy seems to remember that triangular numbers had a common second difference.

Let's have a look.

To do this, I'd like you to write down the first five triangular numbers and see if Izzy is correct.

Let's have a look.

We've got one, three, six, 10, 15.

We're adding two, then three, then four, then five.

So, yeah, we've got a common second difference of plus one.

Any sequence which has a common second difference is called a quadratic sequence.

You don't need to remember the name of this type of sequence yet, but it might help us refer to these different sequences.

So which of these sequences have a common second difference? I'm asking, which could be the first five terms in a quadratic sequence? Pause the video, work it out.

Let's check them.

We're adding one, then two, then three, then four.

So, yes, that does have a common second difference.

We're adding two, then four, then six, then eight.

So we've got a common second difference of two.

And the last one, we're adding two, then four, then eight, then 16.

So, no, we have not got a common second difference.

The differences are increasing by two, then by four, and then by eight.

So let's have a look at the sequence which starts two, three, five, eight, 13.

Laura says it cannot be arithmetic or geometric.

Aisha says, "Maybe we should try and see if it's quadratic then." Take your time.

What do you think? Let's have a look then.

Laura is correct.

It is not an arithmetic or a geometric sequence, but we haven't got a common second difference either.

We're adding one, then two, then three, but then five.

The next term in the sequence is 21.

This takes a little bit of spotting.

Pause the video, have a look at the numbers.

Can you spot a rule that tells us how this sequence is generated? Give it a go.

It's a really good feeling, isn't it, once you start spotting something and checking that that pattern works? So you may have spotted that two plus three is five, but then five plus three is eight, then five plus eight is 13, then eight plus 13 is 21.

Doesn't matter at the moment if you did not spot that yourself, but that's another type of rule that you can keep your eye out for as you're exploring sequences.

The next term would then be 13 plus 21, which is 34.

Laura says, "My sequence starts on 10, and each term is generated by adding the two previous terms".

So she's going to use a similar rule, the one we just looked at.

Why are we unable to write Laura's sequence down? We know the first term and we know the rule.

So what is the problem at the moment? Right, unlike a lot of our rules, we need two terms to start with, 'cause we need to be able to add two things together to get the next term.

So if the second term was four, we've now got enough information to write down this sequence.

We'll do it together.

We've got 10, then four.

Adding those makes 14, adding those makes 18, adding those makes 32.

A little bit of a challenge for you.

Can you change that second term so it's not four anymore? Because I want the third term to be smaller than the first.

Have a play around.

Right, so the second term needs to be negative.

So if we had 10 negative two, we'd then get eight, then six, then 14, then 20, and then our sequence is going to increase again.

If we have something like negative 12, we've got 10, then negative 12, that takes us to negative two, then negative 14, then negative 16, and then that's going to keep decreasing now as we're adding two negative values together.

So have a go yourself for these sequences.

Each term is generated by adding the two previous terms, just like the patterns we've previously looked at.

I would like you to work out the next three terms in each sequence.

Off you go.

Well done.

So you should have three, five, and eight, nine, 17, 26, 13, 18, 31, and two, negative one, one.

If you were going to carry that sequence on, negative one add one is zero, one add zero is one, and then zero add one is one, and then one add one is two.

So it'll take a little while for that one to start increasing again.

Fantastic.

Time to bring all those interesting sequences together.

I'd like you to see if you can work out a pattern for each of these sequences.

Once you think you've got a pattern, find the next two terms in each sequence.

Think about all those different types of sequences we've looked at so far to help you.

Off you go.

Aisha, Izzy, and Lucas have all created a sequence starting with the terms 1/6, 1/2.

Have a read of their sequences, and for each of these questions, I'd like you to justify your answer.

So explain to somebody why you've picked that one as an arithmetic sequence, or a geometric sequence, or a sequence with a common second difference, or that quadratic sequence if you want to remember that new word.

Off you go and come back for the answers.

Well done.

Let's have a look at these answers.

It may have taken you a little time to find the right pattern.

That's absolutely fine.

You had lots of different patterns to check.

Have a read of each of the next two terms. I'm going to explain the rules that I went with.

So for a, I said it was a geometric sequence with a common ratio times two, b, an arithmetic sequence with a common difference plus six, c, a quadratic sequence with a common second difference of plus one.

So it's that one with a common second difference.

You could have said that the differences are increasing by one each time.

D, again, we've got a common second difference of plus two.

The differences are increasing by two each time.

E, if you add the two previous terms, you get the next term.

F, again, seems to have a common second difference, but of plus three this time.

The differences are increasing by three each time.

G, if you add the two previous terms, you get the next term.

That one particularly interesting, because there's not that many sequences that decrease and then start increasing again.

There are a few different types, but it's worth being aware of.

And h, a linear sequence with a common difference of negative six.

Looking at the right hand side, so again, we've got one with a common second difference of plus four.

So that's a quadratic sequence, if you're remembering that keyword.

J, again, we've got a common second difference of plus four.

K, a common second difference this time of plus eight.

That was quite hard to spot.

You need to make sure you draw your arrows on, work out your differences, and then see if you can see the difference in the differences.

L, if you add the two previous terms, you get the next term.

Quite easy to mistake that one for quadratic as it's subtracting two, then three, but then five instead of four.

So well done if you didn't fall into that pitfall.

M, an arithmetic sequence with common difference negative two.

N, geometric sequence, we're multiplying by negative three each time.

And o, this one was really tricky to spot.

So we're subtracting eight, then subtracting three, then adding two, then adding seven, then adding 12.

Just looking at those, it's not always easy to see a pattern.

So if we apply our rule of checking for a common second difference, from negative eight to negative three, we add five, from negative three to two, we add five, from two to seven, we add five, and so on.

So it does have a common second difference of add five, even though we start by subtracting and then we go to adding.

So that is another sequence where we can decrease for a while and then start increasing again.

And finally, well done if you spotted that Izzy's had a common difference of plus 1/3.

Remember, I wanted you to justify your answer.

B, Lucas', it has a common ratio of multiply by three, and then Aisha's, if you change them all over a common denominator, you can see that we're adding 1/3, then 2/3, then 3/3, and so on.

So we have a common second difference of plus 1/3.

Fantastic, I'm hoping that you feel even more experienced now of all the different types of sequences that you could come across.

There's even more types of sequences out there, but it's good to have an idea of what sort of patterns you could be looking for.

So today we've seen that arithmetic sequences can be identified by checking for a common difference between the terms. We're experts now at checking common ratios between terms to see if they're geometric sequences.

We've looked at these interesting sequences that have a common second difference.

We've seen square and triangle numbers following that pattern, and we know that they have this name quadratic sequence, but we don't need to be remembering that one at the moment.

And then there's other rules like that rule we looked at where we added the two previous terms to get the next one.

And there's even more rules out there, which I hope you have some time to explore as you move through with your algebra skills.

Thanks for joining us today.

Really glad that you chose to work with us, and I hope to see you again.