Lesson video

Hello, Mr. Robson here.

Welcome to Maths.

Great choice to join me, especially because we're looking at geometric sequences today and they are wonderful.

I think you'll like this lesson.

Let's get started.

Our learning outcome is I'll be able to recognise the features of a geometric sequence and continue it.

Lots of interesting keywords in our learning today.

A geometric sequence is a sequence with a constant multiplicative relationship between successive terms. A common ratio is a key feature of a geometric sequence.

The constant multiplier between successive terms is called the common ratio.

Look out for that language throughout our lesson.

Two parts to our learning today.

Let's begin by identifying and generating geometric sequences.

Let's get you thinking to start.

Spotting patterns is an essential skill in this world, so can you spot patterns and find the next term in each of these sequences? 3 sequences to ponder.

I'd like you to pause, see if you can spot that next term.

Welcome back.

I wonder what you spotted.

I'm sure you spotted this first one.

A common additive difference between successive terms. That makes this an arithmetic sequence, also called a linear sequence.

If that pattern continues, that next term would be 30.

How about that second sequence? The additive difference, well, it's not a common additive difference.

Or is it? Oh, look, a common second difference.

So we know that is a quadratic sequence.

If this pattern continues, the next term is 80.

The sequence at the bottom, however, there's no common additive difference, but there is a pattern.

Did you spot it? That next term would be 160.

Let's explore this sequence further.

What happens when we look for a common second difference? Pause and see if you can identify the second difference in this sequence.

Welcome back.

Did you see it? Wasn't it lovely? We get those same values again.

What do you think's gonna happen when we look for a third difference? You are ahead of me here, it's that same pattern again.

This beautiful pattern continues on and on.

This is clearly a very special type of sequence.

Andeep and Izzy are trying to classify this sequence.

Andeep says, "I like this one.

I see it as an add itself sequence." And Andeep shows that the terms are generated by doing 5 + 5, 10 + 10, 20 + 20, and so on.

Izzy says, "I think I see a more efficient way to describe this and it's not additive." Can you spot what Izzy has spotted? Pause, tell the person next to you or say it aloud to yourself.

What's Izzy noticed? Welcome back.

Did you see what Izzy had spotted? Izzy says, "It's actually a multiplicative sequence." It's two lots of five, two lots of 10, two lots of 20 and so on.

Andeep says, "That's genius Izzy! We now have a common thing happening from term to term." Little bit of work for you to do now, I'd like you to use Izzy's term to term rule to find the next 3 terms in this sequence.

Pause and do that now.

Welcome back.

The next term after 160 is 320.

Multiply that one by two, 640, multiply that term by two, 1,280.

Sequences with a common multiplicative relationship between successive terms are known as geometric sequences.

That is a really important sentence.

You'll want to pause and write it down.

Quick check you've got that, I'd like you to complete the sentence and find the next five terms of this sequence.

Pause and do this now.

Welcome back.

We had a word missing from that sentence.

I do hope you wrote multiplicative.

Geometric sequences have a common multiplicative relationship between successive terms. So, in this sequence, X 3, X 3, X 3, X 3.

That's a common multiplicative relationship.

This is a geometric sequence.

Let's generate the next five terms: 18 X 3, 54, X that by 3, 162, X that by 3, 486, X that by 3, 1,458, X that by 3, 4,374.

That is a geometric sequence.

Another quick check, which sequence is geometric? Once you've selected which one is geometric, I'd like you to explain how you know that one is geometric and the other two are not.

Pause and do this now.

Welcome back.

I hope you said A is geometric, B and C are not.

But I don't want you to stop there.

I asked you to explain how you know.

How do you know that A is geometric? It's a common multiplicative relationship.

Why is B not geometric? Because it's a common additive relationship, that makes it arithmetic.

You might call it linear, but it's not geometric.

What about C? There's a multiplicative relationship between the terms. Surely that one's geometric.

No, it's not a common multiplicative relationship.

It has to be a common multiplier in order to be geometric.

It's possible to generate geometric sequences if we know just the first term and the common multiplicative relationship.

For example, if I say a first term of 3 and a common multiplicative relationship of X 4, you can generate the sequence.

We'll start at 3 and we'll acknowledge there's gonna be a common multiplier of 4 between terms and then we'll perform those multiplications.

3 X 4, 12 X 4, 48 X 4, 192 X 4, and we've generated that geometric sequence.

I generated that sequence, first term of 3, common multiplicative relationship of 4.

I'd like you to take a turn and generate this sequence.

I'd like the first five terms of the geometric sequence, with the first term 2, and a common multiplicative relationship of X 5.

Pause and work out those five terms now.

Welcome back.

Hopefully you started with 2 and then set about applying that common multiplicative relationship.

2 X 5, 10 X 5, 50 X 5, 250 X 5.

So this geometric sequence goes 2, 10, 50, 250, 1,250.

Lucas and Laura are discussing the common multiplier of geometric sequences.

Lucas says, "From what I've seen, I think the common multiplier between the terms has to be a whole number." Laura says, "I disagree.

I can generate a geometric sequence with a common multiplicative relationship of X 1/2 between terms. Only one of our two pupils is right, but who is it? With whom do you agree? Pause, tell a person next to you or say it aloud to yourself.

Welcome back.

I hope you agreed with Laura.

Let's look at this sequence.

4, 2, 1, 1/2, 1/4.

Luca says, "Nice work, Laura! So both the multiplier and the terms can be non-integers." Well done, Lucas, that's absolutely true for geometric sequences.

Something else that's true is, if the multiplier is between 0 and 1, the sequence decreases.

Quick check you've got that, I'd like you to generate the next four terms of these geometric sequences.

Pause and do that now.

Welcome back.

Hopefully that first sequence read 16, 4, 1, 1/4, 1/16.

It's nice and efficient to write the fourth and fifth terms there as fractions.

For the next sequence, 170, 17, 1.

7, 0.

17, 0.

017.

I wrote those as decimals, you are welcome to write them as fractions, either one is perfectly valid.

And remember, if the multiplier is between 0 and 1, the sequence decreases.

Here's two examples of decreasing geometric sequences.

Practise time now.

Question one, I'd like you to classify these sequences.

Only two things we're gonna classify them into, they're geometric or they're not geometric.

Pause, get classifying.

Question two, I'd like you to generate the first five terms of these geometric sequences.

Pause and do that now.

Question three, I'd like the next four terms of these geometric sequences.

Pause and find them now.

Question four, these ones look a little trickier, but trust me, they're not.

Be brave, give them a go.

I'd like you to generate the next four terms of these geometric sequences.

Feedback time now.

Let's see how we got on with question one.

We were classifying these sequences as geometric or not geometric.

A is absolutely not geometric.

B is geometric, common multiplicative relationship of X 4, you might have noticed.

C was geometric, X 2 between successive terms. D was not geometric.

E was not geometric.

F, well done, geometric, multiplying term to term by 10.

G was not geometric.

H is unusual, it's geometric and we start at -1 and we multiply by 2 to get from term to term.

That's the sequence we get.

Question two, we're generating the first five terms of these geometric sequences.

For A, your sequence should have read, 5, 15, 45, 135, 405.

For B, your sequence should have read, 3, 30, 300, 3000, 30,000.

For C, first term of 10 and a common multiplier of 3, well, that's the same digits that was in part B.

Surely this is the same sequence.

Absolutely not.

For C, the sequence goes, 10, 30, 90, 270, 810.

For D, first term of 1 and a common multiplier of -2.

Well this is interesting.

Your sequence should have gone, one, <v ->2,</v> positive 4, <v ->8,</v> positive 16.

What a beautiful sequence.

Something else to note here, if the multiplier is negative, the sequence alternates.

In that sequence we're alternating from positive term value, to negative term value, to positive term value, and so on.

How nice.

Question three, I asked you to generate the next four terms of these geometric sequences.

The sequence in A should have read, 9, 3, 1, 1/3 1/9.

For B, it should have read, 75, 15, 3, 3/5, 3/25.

It's really efficient to express all those terms as fractions.

Question four, generate the next four terms of these geometric sequences.

Part A, if I take root 2 and multiply it by 5, I get 5 lots of root 2.

When I take five lots of root 2 and multiply it by 5, I get 25 lots of root two, so what's coming next? Well, logically that's going to be 125 lots of root 2, followed by 625 lots of root 2.

For B, five pi multiplied by two, I'll end up with 10 pi and then 20 pi, 40 pi, 80 pi.

How about C? A multiplicative relationship of X pi? Well, 2 pi X pi, is 2 pi squared.

2 pi squared X pi is 2 pi cubed and you start to see this beautiful pattern.

Lovely.

Onto the second half of our learning now, where we're going to look at finding and using the common ratio.

You can find the common multiplicative relationship in a geometric sequence by dividing any term by the term directly before it.

What does that mean? Well, let's look at this geometric sequence.

2, 6, 18, 54, 162.

I'll take the fifth term and divide it by the fourth term.

I'll take the fourth term and divide it by the third term.

I'll take the third term and divide it by the second term.

What do you think I'm gonna do next? Well done.

I'm gonna take the second term and divide it by the first term.

Have you noticed something? Of course you have.

Every time we get a common result of three.

The common multiplicative relationship from term to term is multiply by three.

In a geometric sequence, we call this common multiplicative relationship the common ratio.

That's a really important sentence, so I'd like you to pause and write that down.

Thanks for doing that.

This geometric sequence has a common ratio of 3.

We don't need to say it's got a common multiplicative relationship of X 3.

We don't have to say the term to term rule is X 3 each time, we can just say, nice and concisely, the common ratio is 3.

Quick check you've got this.

I'd like you to identify the common ratio in this geometric sequence.

Pause and try it now.

Welcome back, let's see how we did.

Hopefully you divided the fifth term by the fourth term, the fourth term by the third term, the third term by the second term, and the second term by the first term, and you noticed a common result.

Every time we get a common result of four, therefore the common ratio is four.

We can use a similar strategy to show why something is not a geometric sequence.

2, 6, 24, 72, 288.

It looks like it might be geometric, but let's take a closer look.

When we do the fifth term divided by the fourth term, we get four.

But when I do the fourth term divided by the third term, I get 3.

When we keep dividing each term by the previous term, we notice there's a beautiful pattern to this sequence, but there's no common ratio.

If there's no common ratio between successive terms, it's not a geometric sequence.

Quick check you've got that.

I'd like you to show why this is not a geometric sequence.

Feel free to use a calculator.

Pause and get showing why this one's not geometric.

Welcome back.

Hopefully you divided the fifth term by the fourth term, the fourth term by the third term, the third term by the second term, and the second term by the first term, and you found there's a beautiful pattern, but there's no common ratio between successive terms. Therefore, this is not a geometric sequence.

Well done.

We can use just a handful of terms in order to continue a geometric sequence.

For this sequence, we only know the first three terms, but when we take the third term and divide it by the second term, we get 2.

5, take the second term, divide it by the first term we get 2.

5.

We're told this is a geometric sequence, so the common ratio of 2.

5 will continue.

600 multiplied by 2.

5, gives us the next term, multiply that by 2.

5 and get the next term.

What I'd like you to do is find the next two terms of this geometric sequence.

Pause and try it now.

Welcome back.

Hopefully the first thing you did was identify the common ratio.

You could have done that by dividing the third term by the second term or the second term by the first term.

Whichever one you do, you spot a common ratio of -2.

You can continue the pattern using that common ratio.

If we know it is a geometric sequence, we don't need to know the first terms in order to identify the sequence.

For example, if we know the third and fourth terms of a geometric sequence, can we find the first, second, and fifth terms? If you are feeling brave enough, pause and see if you can figure out those missing terms. Alternatively, hang on and I'll talk you through it.

Step one is we wanna know the common ratio.

We can take 75 and divide it by 15.

That's the fourth term divided by the third term.

That'll tell us we've got a common ratio of five.

To go backwards in the sequence, we'll need to do the inverse of X 5, so I'm gonna take 15 and divide it by 5, and I've found the second term, it's 3.

3 divided by 5 is 3 divided by 5, or 3 over 5, 3/5.

That's the first term.

To find that fifth term, I'm moving forwards in the sequence from 75.

75 multiplied by 5 is 375.

Your turn now.

We know the second and third terms of this geometric sequence.

I'd like you to find the first, fourth and fifth.

Pause and do this now.

Welcome back.

612 divided by 102 will tell you the common ratio.

To go backwards in the sequence, we need to do the inverse of X 6.

102 divided by 6 is 17, that first term, 17.

Then we can move forwards from 612 and from 3,672 we get to 22,032.

It's also possible to find missing terms in geometric sequences when you aren't given successive terms. We don't know the first term of this one, we do know the second term, we don't know the third one, we do know the fourth, we don't know the fifth one.

This is interesting.

John says, "8 ÷ 2 = 4, so the common ratio must be 4." John's not right.

How do you know? Pause and have a think about that.

Welcome back.

How do you know? Well, if it was a common ratio of 4, then the third term would have to be 2 X 4, which would be 8.

If the third term was 8, the fourth term would be 32, but the fourth term is not 32.

So the common ratio can't be four.

It's not as simple as taking 8 and dividing it by 2 to find the common ratio of this sequence.

Jim says, "I checked.

The common ratio can't be 4.

It must be something that multiplies by itself twice to make 4." Well done Jim.

A nice way to think about this is to consider the common ratio to be r.

2 multiplied by r will give us our third term.

When we take that third term and multiply it by r again, we'll get to 8.

So, 2 X r X r will give us 8.

I know you are looking at that and going, I really want to see that simplified.

Well, I'm going to simplify it this far, 2 X r squared = 8, because from there it's really easy to see that r squared = 4.

We're very close to finding the common ratio now.

Jim's jumped in.

"It's 2.

When you square 2, you get 4.

R must be 2.

If we apply a common ratio of 2 to this sequence, we find these missing terms. John is very excited about solving this problem.

He takes his word to Izzy and says, "I'm so proud of myself." And you should be.

That is something to be proud of.

But Izzy says, "Don't celebrate too soon.

There are actually two solutions to this problem." Well, that's interesting.

Two solutions, two different sequences.

How can that be? If we look at the problem again.

And then think again.

June thinks something else which squares to make four.

Can you spot it? June can.

Got it.

<v ->2.

</v> If we know that R squared equals 4, R will be positive 2 or it could be -2.

So in this case our common ratio might be positive 2 or it might be -2.

If it's -2, our sequence is going to look a little different.

Those missing terms would be -1, -4 and -16.

"Well done, Jun, you found both possible sequences." I'd echo Izzy's praise.

That was marvellous maths.

Quick check you've got that.

Here's a geometric sequence, but we're missing some terms. What could the common ratio be for this geometric sequence? Pause.

Have a think about this problem.

Welcome back.

I wonder what you said.

Hopefully you said A.

Absolutely could be 3.

We'd have those missing terms. I hope you also said it could be B.

We might have a common ratio of -3, in which case the sequence would go like so.

However, it could not be option C.

It definitely wasn't 9.

Practise time now.

Question one.

I'd like you to find the common ratio of each of these geometric sequences.

I don't want to know any other terms. Just want to know the common ratio.

Pause and try these now.

Question two.

I'd like you to determine whether each of these could be geometric sequences or not.

Pause.

Have a think about these three sequences.

Question three.

We've got some geometric sequences, but we're missing some terms. Can you find them? Pause.

Give each one of these a go.

Feedback time.

Question one.

We were looking for the common ratio of each of these geometric sequences.

For A we had a common ratio of five.

For B, a common ratio of six.

For C, a common ratio of 3.

5.

And for D, a common ratio of three quarters.

For question two, we were determining whether these sequences could be geometric or not.

A was no common ratio, therefore it's not geometric.

It's a lovely pattern there, but that's not a geometric sequence.

For B, so close yet so far.

No common ratio, therefore not geometric.

For C, we've got a common ratio, just possibly a geometric sequence, if the sequence continues like so.

Question three, finding the missing terms in these geometric sequences.

For part A, we know between 32 and 64, we've got a common ratio of two.

We can use that to populate the rest of the sequence, getting 4, 8, 16 to be followed by 32 and 64.

How about this next sequence B? Well, to get from 117 to 351, we have to multiply by 3.

As a geometric sequence, must be common ratio of 3, and we can use that to find those two missing terms. C was an interesting one.

We haven't got successive terms, so we could identify that R squared equals four.

3 multiplied R multiplied by R makes 12.

We could form the equation 3R squared = 12 from there.

R squared = 4.

And we've been here before.

If R squared = 4, then R could be positive 2 or it could be -2.

So there's two solutions here.

If R is positive 2, that's the sequence.

If R is -2, that's the sequence.

There were two solutions there.

How about D? Well, this is tricky.

We only know the first term and the fourth term, so 7 multiplied by R, multiplied by R, multiplied by R gives us 875.

We form an equation from there, we find that 7R cubed = 875, so R cubed must be 125.

How many solutions to that? Just the one.

R=5.

Common ratio of five will give us those missing terms. We're at the end of the lesson now, sadly, but we've learned that a sequence might be geometric if there's a common multiplicative relationship between successive terms. We call this multiplier the common ratio.

If we know two terms and their position in a geometric sequence, we can determine other terms in the sequence.

Hope you enjoyed this lesson as much as I did and I look forward to seeing you again soon for more mathematics.

Goodbye for now.