Lesson video

Hello, Mr. Robson here.

Superb choice to join me for maths today.

We're extending thinking about sequences in this lesson, and I know you love sequences, so let's get going.

Our learning outcome is that we'll be able to appreciate that abstract sequences can have negative values.

Some key words that are gonna come up throughout today's lesson.

Arithmetic, linear.

An arithmetic, also known as a linear sequence, is a sequence where the difference between successive terms is a constant.

For example, 10, 12, 14, 16, 18.

This is an arithmetic sequence with a common difference of positive two, whereas 10, 12, 15, 19, 24, this is not an arithmetic sequence because there is not a common difference between successive terms. Geometric is another kind of sequence we'll see.

A geometric sequence is a sequence with a constant multiplicative relationship between successive terms. Look out for those keywords throughout today's lesson.

Two parts to our lesson, and we're gonna begin by looking at concrete versus abstract sequences.

There are lots of concrete examples of sequences in the world around us.

For example, tables in a dining hall.

There's one table with some people around it, two tables, some people, three tables and some people.

Can you see a pattern in this sequence? Pause, tell the person next to you or say it aloud to me at the screen.

Welcome back, I wonder what you spotted.

Did you notice one table, 12 people, two tables, 16 people, three tables, 20 people.

IE, every time we had a table, we add four more people.

Well done, you can see the four more people being added at the end of every additional table.

Spotting patterns in sequences enables us to extrapolate future terms. For example, we'd quite like to know how many people are around four tables, five tables, six tables, seven tables, eight tables, et cetera.

But we don't want to draw that diagram every time.

It would take a lot of time and effort.

If we extrapolate that pattern of add four between successive terms, we can see in our table that eight tables will have 40 people seated around them.

Spotting further patterns and relationships enables us to extrapolate terms that are way into the future.

There's a relationship between the number of tables and the number of people.

Take the number of tables, multiply it by four, add eight and you get the number of people.

We could use this algebraic equation, 4T plus eight equals P to express that.

I'm just gonna prove to you that that's a truth, that that equation works.

One table gives us 12 people.

There's an equation.

Two tables gives us 16 people.

The equation works, eight tables gives us 40 people.

The equation works, so can you use that equation, 4T plus eight equals P to tell me how many people are around 100 tables? Pause and work that out.

Welcome back, how did we do? Four lots of 100 plus eight, 408.

There are 408 people seated around 100 tables.

We could extrapolate terms that are way into the future, so not just 100 tables.

We could do 101, 102.

We could skip a few and get to 1000 tables with 4,008 people and from there, 1,001 tables, 1002 tables.

We could skip a few, a million tables, 4,000,008 people, 1,000,001 tables 4,000,012 people, 1,000,002 tables, 4,000,016 people.

It's far easier to use our algebraic equation than it is to continue that sequence by hand or by drawing it.

Using our equation enables us to find large terms efficiently.

How about this large term? How many people around a billion tables? Well done, we'd do four lots of a billion and add eight and we'd get 4,000,000,008.

How many people around two billion tables? Well done, four lots of two billion plus eight gives us 8,000,000,008 people.

What a lovely number that is, but there's a problem with it.

What is the problem with these numbers? Pause, make a suggestion to the person next to you or a suggestion to me at the screen.

See you in a moment.

Welcome back, did you spot it? We've exceeded the population of planet earth.

We don't have 8,000,000,008 people.

Concrete models are often restricted by physical limits.

Quite often that physical limit in this context might be the size of your dining hall or the number of tables you have or the number of people who are coming to dine, but if we run this infinitely, we'd run out of people on planet earth.

So it's important we're aware of this.

Concrete models are often restricted by physical limits.

Quick check you've got that.

True or false, all concrete models can continue infinitely.

Is that true or is it false? And I'd like you to justify your answer with one of these two statements.

Concrete models are often restricted by physical limits or patterns go on infinitely.

Pause, have a get this problem now.

Welcome back, I hope you told me that statement is false and you justify it with the statement, concrete models are often restricted by physical limits.

Let's look at another concrete example.

A farmer needs to raise some money by selling off some land.

The farmer begins with land 10 kilometres by five kilometres in a rectangle, therefore an area of 50 kilometres squared, after one year, they've lost one kilometre off each length of their land.

After year two, they've lost another kilometre of each length of their land.

So our farmer starts with 50 kilometres squared, then has 36 kilometres squared, then has 24 kilometres squared.

Can you spot a pattern in this sequence? Can you find the next two terms? Pause, have a think about this problem now.

Welcome back, lots of ways you might spot this pattern.

I quite like to think of it like this.

We had 10 by five, then nine by four, then eight by three.

So naturally what's coming next? Seven by two and six by one.

So the farmer will have 14 kilometres squared and then six kilometres squared.

What is the problem beyond this? Pause, make a suggestion to the person next to you or to me at the screen, what problem are we about to come up against? See you in a moment.

Welcome back, lots of things you might have said, we're about to run out of land.

If the next dimension is five kilometres by zero kilometres, we've got zero kilometres squared.

But what comes after this four kilometres by negative one kilometres, that can't happen In the concrete world, we can't have a negative area, so the pattern cannot continue like this.

When we model it with a concrete example, sometimes in math though, we want to and can make abstract models.

If I look at that very same sequence, the sequence that goes 50, 36, 24, 14, six and allow zero and negative four, we can start to model abstractly.

Rather than thinking of this as a physical model of dimensions and area, let's spot another pattern.

You look at the additive difference between terms and you'll notice there's a common second difference.

We could use this to extrapolate future terms in our sequence and the sequence will continue, negative six, negative six, negative four, zero, six, and we're back into the positives.

Our abstract sequence can have negative terms and there's times in mathematics when we want to be able to model like this.

Let's check you've got that, true or false? Abstract sequences can have negative terms. Is that true or is it false? I like to justify your answer.

One of these two statements.

The first five terms of this sequence are positive.

The sequence will stop when it reaches zero, or if this sequence is abstract, the terms could become negative.

Pause, decide whether that's true or false.

Don't forget to justify your answer.

See you in a moment.

Welcome back, I hope you said it's true.

Abstract sequences can have negative terms because if this sequence is abstract, the terms could become negative.

They could continue after 19 with 11, three, negative five, negative 13 if you spotted the pattern in that sequence.

Practise time now, for this question, we have a table showing us the history of men's 100 metre sprint world record times, and we've got the data from 1940, 1960, 1980, 2000, 2020.

I'd to identify a pattern and use it to predict the times up to the 2100.

For part B, you know this is a concrete sequence, so what is its physical limit? Write a sentence to answer that one, pause and do this now.

Welcome back, time for some feedback.

For part A, I asked you to identify a pattern and use it to predict the times up to the year 2100.

You should have identified that over the course of 20 years, we lose 0.

16 seconds of the time.

You could say this in arithmetic sequence with a term to term difference of negative 0.

16.

If we extrapolate this into the future, we can see these times for 2040, 2060, 2080 and 2100.

So by the year, 2100, the men's 100 metre sprint world record time would be 8.

94 seconds if this pattern continues, but this is a concrete sequence.

What is its physical limits? There are lots of things we might have said here.

You might have said zero.

We'll never run it in zero seconds, but you might have said, four seconds, five seconds.

There's got to be a limit to how quickly a man can run 100 metres.

I answered it with in the year 3140, we'd see a time of negative 0.

1 seconds.

It would be impossible to have a negative term in this context.

The sequence would become abstract.

Onto the second part of our lesson now, extending sequences in both directions.

This is a geometric sequence with a common ratio of two, but what are the missing first and sixth terms? Pause and work those out for me.

Welcome back.

With a common ratio of two, we might write that underneath our table like so.

We can take 272, multiply it by two to get the sixth term, 544, for the first term, we need to move backwards in this sequence, we'd need to do 34 divided by two to get 17.

You can generate previous terms by using the inverse of the term to term rule.

Sometimes in mathematical modelling, we don't want to limit a sequence to having a first term.

We want to be able to look back beyond it.

A classic example of this is looking back in history, scientists are recording the growth of a bacteria and they get these results.

They notice that the growth fits the model of a geometric sequence.

If you look at that table for the number of bacteria we have week by week, it forms a geometric sequence with a common ratio of 1.

5 between successive terms. We can use this recorded data, make the assumption that this pattern of growth has always existed and look back in history.

There's the assumption that we've always had this growth of 1.

5.

Between successive terms, we can do 12.

96 divided by 1.

5 and get 8.

64, divide again by 1.

5 to get 5.

76, divide again by 1.

5 to get 3.

84.

It might be that we want to know when the bacteria numbered less than 4,000.

We had to go back beyond the first term and we can make the assumption that three weeks before we started recording this data, the bacteria numbered less than 4,000.

This is a really useful example of where we use modelling in the real world.

Quick check you've got that, scientists are recording the radioactivity of a sample.

The data they record forms a geometric sequence.

Make the assumption that this pattern existed previously and find the two terms before the first recorded term.

My hint, when we're looking at geometric sequences, divide each term by the previous term to find the common ratio.

Pause, see if you can find those two terms. See you in a moment.

Welcome back, our first step, we wanna find the common ratio.

Say if I divide the second hour by the first hour, I get 0.

8, the third hour by the second hour, 0.

8, the fourth hour by the third hour, 0.

8, we've got a common ratio of 0.

8.

If I lay on my table like that, that's me assuming that this pattern existed prior to when we started recording the data, and I can then do the inverse to find the two terms that occurred before we started recording.

600,000 divided by 0.

8 is 750,000.

750,000 divided by 0.

8 is 937,500.

That's our estimate for the number of nuclei two hours before we started recording.

Practise time now, scientists start studying the penguin population on an island in the southern ocean annually, and they find these numbers.

The population data forms a geometric sequence.

For part A, I'd like to move forwards to estimate the population next year and in two years time.

For part B, I'd like you to move backwards to find out how many years ago the population was under 100,000.

Pause and do this now.

Once again, you might want your calculators.

Welcome back.

Let's see how we did.

I ask you to move forward for part A to estimate the population next year and in two years time, we need to know the common ratio in this geometric sequence.

200 divided by 160 is 1.

25 and I can check that works for all the terms we know.

So moving forward, I need to multiply our fourth year by 1.

25, which will give me 390.

625 and then multiply that by 1.

25 to get 488.

281.

I'm gonna round those to one decimal place before I put them in the table so we can estimate that fifth year's population to be 390,600, the sixth years population to be 488,300.

Moving backwards for part B to find out how many years ago the population was under 100,000.

We'll make the assumption that this model worked historically, we've always had this common ratio and then we'll do the inverse.

160 divided by 1.

25, 128 divided by 1.

25 and 102.

4 divided by 1.

25.

The population was under 100,000 three years before this study began.

That's end a lesson now, unfortunately, but in summary, we've learned that sequences can apply in the modelling of many situations.

We can use those models to move forward some backwards and make predictions of the future and of the past.

Some concrete models have physical limits.

By mathematics, we can model abstract sequences and they may be infinite as well as having negative values.

Hope you've enjoyed this lesson.

I hope to see you again soon for more mathematics.

Good bye for now.