Lesson video

Hello, Mr. Robson here.

Welcome to Maths, great choice to be here.

Features of geometric sequences.

Well, that sounds like an awesome lesson, so let's not waste any time.

Let's find out what it's all about.

Our learning outcome is that we'll be able to appreciate the features of a geometric sequence.

Key words that we'll be using throughout this lesson, geometric, a geometric sequence is a sequence with a constant multiplicative relationship between successive terms. Common ratio, 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 me saying geometric sequence, constant multiplicative relationship and common ratio throughout this lesson.

Two parts to today's lesson and we're going to start by identifying geometric sequences.

Let's start with something that should be quite familiar to you.

I'm gonna ask you in a moment to pause and spot patterns in these sequences and find the next three terms, which I have blanked out.

I want you to pause and have a go at that now.

Next three terms of each of these sequences.

See you in a moment.

Welcome back.

Did we spot 7, 12, 17, et cetera.

A common additive difference, a positive 5 from term to term.

Should that continue, the next terms will be 32, 37 and 42.

For the second sequence, a lovely pattern.

7, 17, 27, 37.

Common additive difference, positive 10 between the terms. If that continues, we'll go to 57, 67, 77.

The third one was different or was it the same? It was different because it was a decreasing sequence, but it still had a common additive difference.

We added negative 9 from one term to the next.

If we continue to add negative 9s, we'll get to negative 17, negative 26, negative 35, but even though that last sequence decreases, it still belongs to the same family.

The arithmetic sequence family.

Common additive differences between terms makes these arithmetic sequences.

You also see them called linear sequences, a common additive difference.

What about these sequences? Are these ones arithmetic, common additive difference or is there something else going on for some of them? I'm gonna ask you to pause and explore these sequences, what you think the next term is in each case? See you in a moment.

Welcome back.

The first sequence should have looked very familiar.

It's indeed arithmetic a common additive difference.

Positive 5 from term to term.

A common additive difference makes it a linear sequence.

If we continue that positive 5 pattern, then we get to 30, nice sequence.

What about the second one though? It was different.

How? All of a sudden we didn't have a common additive difference? You could look at the term to term additive difference and it went add 5, add 10, add 15, add 20.

Oh, there's a pattern to what it's adding.

There's no common additive difference, but there's a common second difference.

Sequences with a common second difference.

They're a special kind of non-linear sequence, but assuming that common second difference continues, we'll need to add 25 to 55 to get the next term, which was 80.

That's a nice sequence.

How about that third one? Was it the same as our arithmetic sequence, the same as our sequence with a common second difference or was it different again? Well done, it's different again.

When we look at the differences between the terms, we add 5, then we add 10, then we add 20, then we add 40.

There's no common additive difference, but there's certainly a pattern and you spot it.

We have to add 80 to get the next term.

Of course, the first term's 5, add another 5, the second term 10, add another 10 and so on.

So the sixth term in that sequence was 80, add 80, 160.

Can you spot it? This last sequence is a special type of sequence.

And we look for a common second difference.

Well, we know the first difference.

We go add 5, add 10, add 20, add 40, add 80.

If we look for a difference between those differences, 5, 10, 20, 40, 80, what about the difference in those differences? 5, 10, 20, 40, 80.

Interesting.

This is clearly a very unique type of sequence.

Andeep says, "I like this one.

I see it as an add itself sequence." We do 5, add 5 to get the next term, 10 add 10 to get the next term.

20 add 20 to get the next term.

And you see why Andeep calls it an add itself sequence.

Nice.

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 this video, have a conversation with a person next to you or a good think to yourself.

What has Izzy spotted? See you in a moment.

Welcome back.

Izzy has spotted it's actually a multiplicative sequence.

Instead of 5, add 5, 10, add 10, 20, add 20.

We could call it 5 times 2, 10 times 2, 20 times 2 and so on.

Did you spot that? A more efficient way of expressing any of those additions.

It's more efficient because it really efficiently describes what's going in in our sequence.

5 times 2 makes 10, 10 times 2 makes 20, 20 times 2 makes 40 and so on.

Andeep says, "That's far more efficient Izzy.

Well spotted." It's lovely to see pupils praising other pupils in the classroom.

Let's use Izzy's term-to-term rule to find the next 3 terms. Assuming this pattern continues with the-term-to-term rule, where we will multiply by two.

We're gonna go from 160, multiply that by 2, 320, multiply that by 2, 640, multiply that by 2 1,280.

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

That sentence is important, so I'm going to suggest you pause now and copy it down.

See you in a moment.

Quick check that you've got everything I've said so far.

True or false? Geometric sequences have a prominent additive difference between successive terms. Is that true or is it false? Once you've decided whether it's true or false, could you use either statement A or statement B to justify your answer? Pause.

Have a think about this.

I'll be back in a moment.

Welcome back.

False.

I hope you were false.

Geometric sequences have a common additive difference between successive terms. That's false.

Why is it false? It's false because the common term-to-term rule in a geometric sequence is multiplicative.

Next check.

I'd like to find the next 5 terms of this geometric sequence.

You can pause this video, copy that down and find the next 5 terms. See you in a moment.

Welcome back.

What did we know? We knew it was a geometric sequence.

What do we know about geometric sequences? They have a common multiplicative relationship between successive terms, a common multiplicative relationship, which means that multiply 3 that we've seen between the first three terms will continue.

Therefore 18 multiply by 3 makes 54.

54 multiplied by 3 makes 162.

162 multiplied by 3 makes 486.

It does 486 times 3 makes 1,458.

Yep.

1,458 times 3 makes 4,374.

The last one was a tricky one.

Maybe next time I'll use my calculator for that one.

Last check.

What is the difference between the nature of these two sequences? Pause, spot it.

Tell the person next to you.

Tell me back on screen.

What's the difference? See you in a moment.

Welcome back.

I hope you said the top sequence, 2, 4, 6, 8, 10 has a common additive difference between terms. That makes it a linear or arithmetic sequence, whereas the second sequence has a multiplicative relationship between terms. A common multiplicative relationship between successive terms makes it a geometric sequence.

It's important that you understand the difference between arithmetic sequences and geometric sequences.

Andeep and Izzy are looking at this sequence.

Andeep asks, "This sequence is multiplicative between terms. Does that mean it's geometric?" What do you think? Pause, tell the person next to you or say it aloud to me on screen.

I'll see you in a moment.

Izzy spotted it.

No, it's not geometric.

Be careful.

It has to be the same multiplier between terms. Geometric sequences have a common multiplicative relationship between successive terms. I highlighted the word common there.

It has to be the same multiplier between successive terms. Quick check.

You've got that.

Geometric sequences have blank multiplicative relationship between successive terms. What's going in that blank space? Is it an increasing, a common or an additive? Which of those words are you gonna point that space? Pause.

Tell the person next to you.

See you in a moment.

Welcome back.

It was option B, a common, so the sentence would read geometric sequences have a common multiplicative relationship between successive terms. Practise time now, question one.

I'd like to sort these sequences into geometric, arithmetic and other.

Pause, do so and I'll see you in a moment.

Feedback time.

I've got 3 spaces, one for geometric sequences, one for the arithmetic sequences and one for the other.

Going through them in order sequence A, 1, 2, 3, 4, 5 is an arithmetic sequence, common additive difference term plus 1.

B was geometric.

1, 2, 4, 8, 16 is a common multiplicative relationship between those terms of multiply by 2.

C, well that's in the other category.

We look at the differences.

Add 1, add 3, add 5, add 7.

There's no common additive difference.

It's not arithmetic.

There's no common multiplicative difference, so it's not geometric.

It goes in the other pile.

B was unusual, but it was geometric.

Negative 1, negative 2, negative 4, negative 8, negative 16 looks awfully similar to B, doesn't it? Only we're starting on a negative term and multiplying by 2 from term to term that makes it with a common multiplicative relationship a geometric sequence.

E was also geometric instead of a multiplier of times 2 between terms, we had a common multiplicative relationship of times 4 between terms. F was arithmetic.

4, 16, 28, 40, 52.

Common additive difference of plus 12 between the terms. G was in the other.

There's no common additive difference.

There's no common multiplicative difference between successive terms. G was in the other sequences pile.

H was also in there.

Lovely sequence add 5, add 5.

Almost looks like it's arithmetic, but then it changes to add 10, add 10.

That's not a common additive difference, therefore it's not arithmetic.

I was arithmetic with a common additive difference plus 30 between terms. J is in the other sequences file.

If you look at the multiplicative relationships, it multiplies by 2, then multiplies by 3, then multiplies by 4, then multiplies by 5.

Is that a common multiplicative relationship? No, so it's not geometric K was geometric.

10, 20, 40, 80, 160.

That's a common multiplicative relationship times 2 between the terms. And L also 10, 100, 1000, 10,000.

Common multiplicative relationship times 10 between successive terms that makes it geometric.

You might wanna pause now and just check that you've got all your sequences in the right categories.

On to part two of the lesson, generating geometric sequences.

Lucas and Laura are generating geometric sequences.

Lucas says, "I'm gonna start with a first term of 1 and use a common multiplier of 5 between the terms." Starts on 1 means a common multiplier of 5 and the sequence will go 1, 5, 25, 125.

Can you guess the next one? Well done, 625.

Laura says, "I think I can generate the exact same sequence by starting on 1 and using a ratio of 1 to 5 between the terms." Do you agree with Laura? Do you think she's gonna get the exact same sequence as Lucas? Pause, have a think, have a conversation with the person next to you.

I'll be back in a moment.

Welcome back.

Let's test Laura's idea starting on 1, using a ratio of 1 to 5 between the terms. Will she generate the same sequence? 1, 5, 25, 125, 625.

So the ratio 1 to 5, that's the first and second terms. If we change that second terms to the third term, using the same ratio 1 to 5, the equivalent ratio 5 to 25 does indeed generate that third term of 25.

What if we leap from the third term to the fourth term by the same ratio? Is that the same ratio? 25 to 125 and then the ratio 1 to 5? Yes they are.

And then we can go from 125 to 625 in exactly the same ratio giving us the fourth and fifth terms. So we can use a ratio to describe the multiplicative relationships between successive terms in a geometric sequence.

Using a common multiplier of 5 or a ratio of 1 to 5 between terms generates the same geometric sequence.

The language we use for the common multiplier is to call it the common ratio.

So the sequence that goes 1,5, 25, 125, 625 it has a common ratio of 5.

Lucas says, "Let's check, I've got this Laura, I can start on 7 and use a ratio of 1 to 2 to generate a sequence." Well, let's see Lucas, can you? There's the ratio 1 to 2.

We're starting on the first term of 7.

7 in the ratio one to 2 will give us the ratio 7 to 14 they're equivalent ratios.

We then go from 14 to 28.

14 to 28 is the same ratio as 1 to 2 and then 28 to 56 is in the same ratio as 1 to 2.

They're all equivalent ratios.

"Well done Lucas," Laura says, "The sequence 7, 14, 28, 56 is a geometric sequence with a common ratio of 2.

Lucas is now doing what we love to see mathematicians do.

He's just questioning something.

Is this a thing? Will it work? Could we? Important questions when we're discovering mathematics.

Lucas says, "You couldn't have a common ratio of half though, could you Laura? It has to be greater than one, right?" Can you answer Lucas's question? What do you think? Pause, have a conversation.

Have a think, I'll reveal the answer in a moment.

Welcome back.

I wonder what you thought.

What did your intuition tell you? Common ratio has to be above 1 or can it be less than 1? Laura says it is possible.

For example, start with the first term of 128 and use a ratio of 2 to 1 between the terms. Starting at 128 in the ratio of 2 to 1 will give us 64.

64 in the ratio of 2 to 1 will give us 32.

32 in the ratio of 2 to 1 will give us 16.

16 in the ratio of 2 to 1 will give us 8.

Lucas says, "Nice, the sequence 128, 64, 32, 16, 8.

So using the ratio of 2 to 1, a common ratio of half between the terms." And there it is the ratio of 2 to 1.

The multiplicative relationship, multiply by half makes the sequence decrease.

So, it is possible to have a decreasing geometric sequence if you are on a multiplicative relationship is less than one.

Right, check time now.

I'm gonna generate a sequence following a rule and then I'll ask you to do the same thing.

So my question is generate the first 5 terms of the geometric sequence with a first term of 81 and a ratio of 3 to 4 between successive terms. So I need the ratio 3 to 4 and then need my first term of 81.

So 81 and the ratio of 3 to 4.

Lots of ways you could do this.

I like to think of it as 81 divided by 3 to find the value of one part and then I'll multiply that by 4 to get the next number in the ratio.

That gives me 81 to 108.

108 divide that by 3, multiply by 4 will take me to 144.

144 divided by 3 multiplied by 4 takes me to 192.

I repeat the process to take me to 256.

I'd get the sequence 81, 108, 144, 192, 256.

Can you repeat that skill for the geometric sequence for the first term of 16 and a ratio of 2 to 5 between successive terms? Pause, give that a go.

See if you can generate the first 5 terms. See you in a moment.

Welcome back.

Let's see how we got along.

The ratio of 2 to 5, starting on 16.

Divide that by 2, multiply it by 5.

We get to 40, 40 divided by 2, multiply it by 5 we get to a 100 divide it by 2, multiply it by 5.

We get to 250 and repeat that and we get to 625.

Your sequence should have gone 16, 40, 100, 250, 625.

Another one to check.

You've got this.

I'm gonna generate the first 7 terms of the geometric sequence, the first term, 405 and a common ratio of one third, and I'll ask you to do something similar.

I'm gonna use my calculator for this one.

I'd recommend you do the same.

I'm gonna start by typing in my first term, 405 and pressing equals and my calculator tells me the answer is 405.

Now why do I wanna do that? It's because I'm gonna use my answer button on my calculator to generate my sequence.

405 is my first term.

405 is now the answer in my calculator.

So I ask my calculator to do answer times a third and it gives me 135, which is the second term in my sequence.

When I press equals again, it's gonna substitute 135 into the answer.

So 135 times a third, becomes 45.

That's the next number in my sequence.

I'm just gonna press equals again and it's gonna find one third of 45 for me.

15.

It just keeps going and generating this sequence for me.

So my sequence goes 405, 135, 45, 15, 5, 5 over 3, 5 over 9.

A common ratio of one third between successive terms. With tip view, non-integer terms are often more efficiently and accurately communicated as fractions.

My calculator told me that sixth term was 5 over 3 and it told me the seventh term was 5 over 9.

I'm happy to leave them in fraction form.

It's a really efficient way of writing those numbers.

It's your turn now.

I'd like to generate the first six terms, the first six terms of the geometric sequence for the first term of 375 and a common ratio of one fifth.

Pause, grab your calculator, work out those first six terms. I'll see you in a minute to check out how you got along.

Alright, I'm back.

Back to see how we've done.

Did we start with a first term of 375, a common multiplicative relationship, a common ratio of a fifth between terms? Did we get the terms 375, 75, 15, 3, 3 over 5, 3 over 25.

Did you write your fifth and sixth terms in fraction form? I hope so.

If you wrote 0.

6 or 0.

12, you are not mathematically incorrect.

It's just quicker and easier, more efficient to write in fraction form more often.

Next, Lucas and Laura are generating geometric sequences, are generating a lot of terms in this geometric sequence and Lucas notices something.

"I've continued this geometric sequence and realised something really cool.

Despite decreasing it will never reach zero." And he's right.

Have you spotted a pattern in this sequence? What do you think the next term's going to be? It goes 1 over 2, 1 over 4, 1 over 8, 1 over 16.

The next terms are 1 over 32, 1 over 64, 1 over 128.

No matter how far it goes, those fractions are gonna get really small, but they'll never reach zero.

So well done Lucas, you have recognised something really cool.

Laura asks, "Does that mean we will never see negative terms in geometric sequences?" Can you answer Laura's question? Pause, have a think, have a conversation with the person next to you.

I would reveal the answer in a moment.

Lucas is right on top of this one.

He says, "No.

I saw a sequence with a common ratio of multiply by 2 and a first term of negative 1 earlier.

It had all negative terms." Do you remember that one from the sorting activity? Lucas does, well done Lucas.

Laura now has a brilliant suggestion.

"Nice.

Let's experiment.

What if we switch them around?" the common ratio and the first term.

"What if we make the first term positive 2 and the multiplier, the common ratio negative 1?" The sequence is gonna start on 2.

We've got a common ratio of negative 1.

What do you think is going to happen? Come back.

How did we do? Did we enjoy this geometric sequence? Is there anything unusual about it? The sequence goes 2, negative 2, 2, negative 2, 2, negative 2 and Lucas points out it's oscillating between positive and negative.

That's beautiful.

Lucas I entirely agree it is beautiful.

Quick check.

You've got that now.

Find the next 4 terms of this geometric sequence.

The first term of 3 on a ratio of negative 3.

The second term would be negative 9.

What's the third term, the fourth term, the fifth term, the sixth term? I'll leave you to work those out.

See you in a moment.

Let's see how we did.

Negative 9 multiplied by negative 3 gives us positive 27.

Multiply that by negative 3 we get negative 81.

Multiply that by negative 3 we get positive 243.

Multiply that by negative 3 and we get negative 729.

What a delightful sequence.

When the common ratio is negative, we get oscillating geometric sequences IE they oscillate from positive to negative to positive to negative.

Practise time now.

Question one.

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

Pause.

Give this a go.

For question two, Alex is generating these geometric sequences and Alex says, because these multipliers are fractions, all of these will be decreasing sequences.

What I'd like you to do is show Alex that he is wrong.

I'll leave you to decide how you are going to do that.

Pause, give this a go.

We'll see how we did in a minute.

Feedback time.

Generating the first five terms of these geometric sequences.

For A, we should have got 1, 3, 9, 27, 81.

First term of 1 common ratio of 3, but B looks similar, slight difference.

It's 10, then 30, then 90, then 270, then 810.

First term, 5 ratio 1 to 3 between the terms, that is like a common ratio of 3, so it'll go 5, 15, 40, 135, 405.

Starting on 3 a common ratio of 5 or ratio of 1 to 5 between the terms it give us a sequence 3, 15, 75, 375, 1,875.

Our E, first term of 64, a ratio of 2 to 5 between the terms, that one will go 64, 160, 400, 1000, 2,500.

For part F starting on 81.

Common ratio of a third.

That'll go 81, 27, 9, 3, 1.

Part G.

A ratio of 5 to 1 between the terms is like a common ratio of a fifth, so we'll start at 1,875 and then get to 375, 75, 15, 3.

And for H, first term of 3 and a ratio of 3 to 1 between the terms. That's like a common ratio of a third.

That sequence, we'll go 3, 1, 1 over 3, 1 over 9, 1 over 27.

You might want to pause and just check that your answers match mine and copy down any errors that you made.

For question two, three sequences and Alex said, "Because these multipliers are fractions, all of these will be decreasing." I said, show Alex that he's wrong.

Lots of ways you could have done this.

One way, generate the sequences.

216, 72, 24, 8.

They're the first four terms of the geometric sequence.

First term, 216 common ratio of a third.

If we start 216, common ratio of two thirds, that goes 216, 144, 96, 64.

The C however, a common ratio of 3 over 2.

That sequence goes 216, 324, 486, 729.

Multiplier in sequence C was greater than one.

3 over 2 is like 1 and a half.

It's like 1.

5.

The multiplier is greater than 1, therefore the sequence is increasing.

That's 1 way you might have showed Alex that he's wrong.

Sadly, reached the end of the lesson now.

What have we learned? We learned that geometric sequences have the key feature of a common multiplicative relationship from term to term.

This multiplicative relationship is also called a common ratio.

For example 3, 30, 300, 3000, 30,000 has a common multiplier times 10 between successive terms. We call that a common ratio of 10.

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

Good bye for now.