Lesson video

It's really nice to see you in this lesson today.

My name is Ms. Davies and I'm gonna be helping you as you progress through the lesson.

With that in mind, please feel free to pause things so you can really think about the concepts we are discussing.

Let's get started.

Welcome to today's lesson on arithmetic sequences.

By the end of the lesson, you'll appreciate the features of an arithmetic sequence and be able to recognise one.

Let's get started.

So an arithmetic or linear sequence is our main definition for today.

We're gonna define this as part of the lesson.

So we're gonna look at this in two parts.

We're gonna identify arithmetic sequences and then we're gonna describe arithmetic sequences.

Below are some examples of arithmetic sequences.

2, 4, 6, 8.

5, 10, 15, 20.

10, 14, 18, 22.

13, 11, 9, 7.

We could have 5, -5, -15, -25.

Or 3/5, 2/5, 1/5, 0.

There's lots of examples.

Maybe have a little bit of a look at some of those features.

Here are some sequences that are not arithmetic.

2, 4, 8, 16.

5, 10, 15, 25.

1, 2, 4, 7.

400, 40, 4, 0.

4.

3, 5, 8, 13.

And 0.

8, 0.

3, -0.

3, -0.

8.

What I would like you to do now is pause the video, have a look at those examples and those non-examples, and see if you could come up with what makes something an arithmetic sequence? What do you think an arithmetic sequence must be from those examples and those non-examples? Give it a go and then we'll define it formally together.

Okay, let's look at what we said.

So an arithmetic or linear sequence is a sequence where the difference between successive terms is a constant.

Now that's probably not the way you phrased it, so let's see what that means.

So what that means is the difference between terms that are next to each other stays the same.

So that means they're going up or going down by the same step each time.

So we're either adding on the same value or subtracting the same value each time.

Let's look at 2, 4, 6, 8.

We're adding two, adding two, adding two.

That difference between successive terms is constantly two, it's always two.

Same with 10, 14, 18, 22, but this time we're always adding on four.

We can have a decreasing arithmetic sequence where we're adding negative two each time.

Or we can have a arithmetic sequence using fractions or decimals.

So here we're adding negative 1/5 each time.

Just to add that we can call arithmetic sequences linear sequences, so I'm gonna be using both key words this lesson.

So how can we explain why the sequence is not an arithmetic sequence? Well, let's look at the difference between successive terms. So we've got plus two, plus four, plus eight.

The difference is not constant, therefore it's not an arithmetic sequence.

Let's have a look at 5, 10, 15, 25.

We add 5, we add 5, then we add 10.

So again, we've not got this constant difference, therefore not an arithmetic sequence.

Okay, let's see if you've got your head around what an arithmetic sequence is then.

So have a look at those four sequences.

Which of those are arithmetic? Off you go.

Well done if you spotted that it's the bottom two.

C is a sequence adding negative three each time.

And D is a sequence where we're adding one each time.

Pictorial sequences can form arithmetic sequences too.

Let's have a look at some examples.

So here we've got a sequence where we seem to be adding two shaded squares each time.

So that so far seems to form an arithmetic sequence.

We've got an example here where we seem to be adding four circles each time.

So again, so far that is an arithmetic sequence.

We've got one where we're adding a single red block each time, and one where we're adding four pink blocks each time.

Because we're adding the same value each time, that makes it an arithmetic sequence.

So let's have a look at this pictorial sequence.

And Izzy says, "The number of lines in each pattern is an arithmetic sequence." How could we check if Izzy is correct? What do you think? Yeah, perfect, we can write down the number of lines in each pattern.

So we've got 5, 9, 13.

So at the moment that seems to form an arithmetic sequence, we're adding four each time.

Jacob says, "The perimeter of each pattern will form the same arithmetic sequence." What do you think to this one? Let's explore it together and then we'll think about your answers.

So the perimeter, we've got 5, then 8, then 11.

So again, so far with the three turns we have, that seems to form an arithmetic sequence.

However, it is a different one to the number of lines.

So Jacob was sort of half correct in his statement.

Okay, time for a check.

For each sequence, does the number of lines in each pattern form an arithmetic sequence? Off you go.

Good work, guys.

First one, yes, it is an arithmetic sequence.

We're adding three more lines each time.

B, not an arithmetic sequence, we're adding three, then adding two.

So we have got a pattern in our sequence, but it's not arithmetic.

And the last one, well done if you spotted that the number of lines is again an arithmetic sequence, we're adding two more lines each time.

So what we can do then is if we know a sequence is linear, linear and arithmetic meaning the same thing, we can find missing terms. So here are two terms in a linear sequence but I don't know the term in between them.

The difference between 12 and 18 is 6, so let's draw that on our diagram.

In order to find that missing value, we need to think about what we add on each time.

So if from 12 to 18 we've added on 6, that's actually two additions, isn't it in terms of moving from one term to the next? We're adding something and then adding the same thing again.

So we do six divided by two, that'll tell us what each addition is.

That means we're adding three each time and therefore we can fill in our missing values.

We've got 15, 21, and then doing our inverse, subtracting three, we get to nine.

So Alex says, "This cannot form a linear sequence as nothing can be added to three twice to get six." What do you think about his statement? Good thinking there.

Let's look at this.

So from three to six we are adding three.

Each addition then must be adding 1.

5 'cause there's two additions to get from three to six.

Then we would fill in 4.

5 and 7.

5.

So now Alex is incorrect.

The difference between successive terms can be a fraction or a decimal, that's absolutely fine.

Okay, so Sophia is looking at sequence -5, 5, -5, 5, and says "The difference between each term is 10, so this should be a linear sequence." Sounds like she might have some doubts.

Can you explain why Sophia is wrong? Let's have a look.

So our difference get from -5 to 5 we're adding 10, then we're adding -10, then we're adding 10.

The value being added each time is not the same, so this is not arithmetic.

So we've got 33, two missing terms and then 21.

Alex has given this another go and says, "The difference between 33 and 21 is 12.

This could be a linear sequence where we are subtracting four each time." Sophia says, "I agree this could be a linear sequence, but there are two missing terms, and 12 divided by 2 is 6.

So the sequence subtracts 6 each time." Who do you think is correct? Let's work it out together.

So the difference is 12, we are subtracting 12 to get from 33 to 21.

So Alex looks correct so far.

Then we've got three additions or subtractions to get from 33 to 21.

12 divided by 3 is 4, so we must be subtracting 4 each time.

Alex then is correct.

Let's see where Sophia's gone wrong.

Sophia is correct that there's two missing terms, but it's not two additions, it's three additions in total.

So you've got to make sure that you're dividing by the right value.

Think about how many times you're adding the same thing to you get from one term to the next known term.

Time for a check.

Terms in arithmetic sequences increase in value by the same amount.

Think about whether that's true or false and what the justification would be for your answer.

Well done if you spotted that one is false.

Adding a negative value will mean successive terms decrease.

So arithmetic sequences don't just increase, we've looked at ones already haven't we where they decrease.

An arithmetic sequence can have both positive and negative terms. What do you think to this one? Well done if you said that was true.

If you add a positive number to a negative number enough times, you will get a positive value.

So if you start on a negative value, you'll have a negative term in your sequence.

But then if you add a positive value enough times, remember sequences carry on infinitely, then you will get to a positive value.

So it is possible to have positive and negative values in an arithmetic sequence.

A chance for you to have a practise.

So for each set of terms, I'd like you to work out if they could form an arithmetic sequence.

And then I'd like you to think about how you would explain your answer.

The focus here is on how you would explain whether these could or could not be arithmetic sequences.

Give it a go and then we'll have a look at the next set.

Good start.

Let's have a look at question two.

So for each sequence made out of different rectangles, I'd like you to work out the perimeter and the area.

And I'd like you to think about whether they form a linear sequence or not? So I've given you a table, if you fill in the perimeters and the areas, and then tell me whether the perimeters form a linear sequence or the areas form a linear sequence.

Off you go.

Well done, guys.

Let's look at our answers.

So A, yes, it could be a linear sequence where we add four each time.

B, yes, again, we could have an arithmetic sequence where we add 0.

05 each time.

C, this one didn't work.

To get from 4 to 16, you could have a rule of add 6, but that would make the fifth term 28, not the sixth term.

And then D, yes, we could add negative two each time, or you could have said subtract two each time.

Let's look at E.

So I've decided that it'll be easier to write these with a common denominator.

So I've changed the mixed numbers into improper fractions and then I've gone with a common denominator of six.

So 2 1/2 is 15/6, 2 is 12/6, and 5/3 is 10/6.

And then I can see clearly a pattern.

So yes, this could be an arithmetic sequence, but we're adding negative six each time.

And then finally our perimeters and our areas.

So for A, we've got 8, 10, 12 and that is a linear sequence.

Our areas are 4, 6, 8, and again that is a linear sequence.

So far we only have the first three turns, so far they are linear sequences.

B, perimeter is 8, 12, 20 so that cannot be a linear sequence.

And the areas are 4, 8, 16, again, not a linear sequence.

Looking at C, the perimeter is 8, 12, 16.

So so far we're adding four each time and that is a linear sequence.

Areas are 4, 9, 16, they are not linear sequences.

You might recognise those numbers as all square numbers, two squared, three squared, four squared.

Doesn't form a linear sequence.

D, the perimeters 18, 16, 14 do form a linear sequence so far.

And then the areas 8, 12, and 12 do not form a linear sequence.

Well done, guys, I hope you enjoyed reminding yourself your perimeter and your area skills and linking those to our new sequence skills.

We're now gonna look at describing arithmetic sequences.

So we can describe sequences using key vocab.

It's important that we can describe a specific sequence in a unique way so that if I describe a sequence, you know exactly what sequence I'm talking about, exactly what values are in my sequence.

So I'd like you to have a go at creating some sentences to describe the sequences below.

I've given you some words in the box that you can try and use.

We'll talk about our answers in a moment, so pause the video, you might want to write down how you would describe those sequences.

Good effort, guys.

So here are some examples of things you may have said.

You might not have gone about it the same way, you might have even better explanations than mine.

So for A, you might have said the sequence is increasing by four each time.

You might have said this sequence is decreasing by five each time for the second one.

For C, each term is half the previous term.

And D, the difference between the first and second term is two.

The differences between terms increased by one each time, that was quite hard to explain.

I've tried to talk about how it's increasing by two, then three, then four, then five.

So I've said the differences between terms increase by one each time.

Hopefully what you found when you're writing those statements is that using keywords such as term or difference makes it a lot easier to explain what is happening.

So Lucas has said, "Sequence a is arithmetic because it goes up in fours." How can we improve Lucas' statement? Okay, we could say something like the sequence increases by four each time.

That's a better word than saying goes up by four.

Even better might be to say the difference between successive terms is four.

We know an arithmetic sequence is a sequence where the difference between successive terms is constant.

We often refer to that constant as the common difference.

So we're gonna bring that language in now.

So Lucas could have said, "The sequence is arithmetic because the common difference between successive terms is four." Which other sequence in our list is linear, and can you explain how you know? See if you use that phrase common difference.

So B was the other linear sequence, you could have said, "B is also linear because the common difference between successive terms is negative five." You might not spend lots of time thinking about your language when you do your mathematics.

However, this is so important because it means other people can really follow what we're trying to say.

So taking our time just to develop those key words and using them regularly will really help our mathematics skills.

So we can use the common difference to identify sequences that are not linear.

So let's see if we can explain why this one is not linear.

We subtract 40, then subtract 20, then subtract 10, then subtract 5.

The differences are not the same, so not linear.

You might have noticed there is a pattern with the differences, we know they need to be exactly the same if it's a linear sequence.

Let's look at 1, 3, 6, 10, 15.

Add two, add three, add four, add five.

Right, there's definitely a pattern there, so it definitely is an interesting sequence, but it's not linear because those differences are not the same.

So Lucas has formed his own sequence and he is describing it to his classmates.

"My sequence is arithmetic, successive terms have a common difference of 10." What is good about his description? Lucas is learning now, isn't he? He's given us some really key pieces of information.

He's telling us it's arithmetic, we know then that it must have this constant additive pattern, and he's given us a real clear common difference between terms. However, will Lucas' classmates be able to guess his sequence? What do you think? You'll possibly notice that no, we're still missing a piece of information.

Lots of sequences have a common difference of 10.

We could have 10, 20, 30 or 55, 65, 75.

So what other information do you think Lucas needs to give so that we know his exact sequence? We need to know one of the terms and we need to know where it is in the sequence.

So we need to preferably know the first term, but he could tell us what the third term is and then we can work that out.

So we can uniquely describe a sequence with an initial term and a common difference.

As long as we have those two facts, then our sequence will be unique.

The initial term is just another word for the first term, or we could call it T1 if we're using that notation.

If we're given any term and the common difference, we could calculate the first term.

So Lucas says, "My sequence is arithmetic, the third term is 15 and terms increase by 10 each time." Let's have a go at rewriting Lucas' statement so it includes the initial term and the common difference.

So we know the common difference is 10.

We can then do the inverse, so subtract 10 to get from the third term to the second term, and then to the first term.

So our initial term is negative five.

We can rewrite his statement now then as an arithmetic sequence with initial term negative five and a common difference of 10.

And then it'll be really easy for somebody to write out the first few terms in my sequence.

Being able to describe linear sequences or the initial term and common difference can help us investigate properties of different linear sequences.

So let's have a look at some linear sequences with algebraic terms. We've got b, b plus c, b plus 2c, and b plus 3c.

Let's have a look at our common difference.

So we are adding c each time, therefore yes, it has an initial term of b and a common difference of c, so it is linear.

How about this one? x, x plus b, x plus 3b, and x plus 6b.

Let's investigate it together.

So add b, add 2b, add 3b.

So no, it cannot be a linear sequence 'cause the difference between the terms change.

Right, let's check some of our key words.

So I'd like you to fill in the blanks to describe each of the sequences.

Try and use the key words where possible.

I've written some below to help you.

Off you go.

Fantastic.

So the first one you could have said is an arithmetic sequence with initial term of -3 and a common difference of 11.

B is a linear sequence with an initial term of 12 and a common difference of -3.

And C, we've got T1, T2, -4, 3.

So we're gonna need to work out our initial term.

So this is a linear or an arithmetic sequence with initial term of -18 and a common difference of 7.

Hopefully now you're feeling really confident with how you can describe an arithmetic sequence so that everybody would write the same sequence down.

Time for practise then.

So for each sequence I'd like you to state whether it's arithmetic or not? Then if it is arithmetic, can you describe it? Think about using all those things we've talked about today, like the language of common difference and initial term, okay, to help you.

Second set.

So below are the first four terms in four different linear sequences.

I would like you to fill in the table with true or false for each of those statements.

So to start with, 1, 5, 9, 13.

Does it contain negative numbers, so true or false? 7, 13, 19, 25 contains negative numbers, true or false? And do the same for each sequence and each statement.

Off you go.

Well done on all that hard work, let's have a look at our answers.

So 1a, yes it's arithmetic, it's got an initial term six and a common difference of nine.

B, not arithmetic, it's got a difference of seven and then three.

C, arithmetic, initial term five and common difference negative eight.

D, not arithmetic, we've got a difference of two, and then six, and then eighteen.

You might have noticed that what this sequence is actually doing is it's multiplying by three each time.

Not an arithmetic sequence though.

And then E, not arithmetic, it starts off with a common difference of 1.

1, but then it has a difference of 2.

1, so not arithmetic.

Looking at the right hand side, F is not arithmetic, it's got a difference of negative 10, then negative 5, then negative 5.

G, it is arithmetic, well done if you spotted this one, it's got an initial term of x and a common difference of two.

H, not arithmetic, we're adding 2x, then 4x, then 8x.

That was a tricky one, good spot if you got that one.

I, arithmetic, it's got an initial term of x and then we're adding on an x and a one.

So you can write that as adding on x plus one.

And then J, it's arithmetic again, initial term of x subtract two, and a common difference of two.

Well done for all that hard work.

Let's have a look at that second set together.

So I'm gonna talk about them in rows.

So containing negative numbers.

So we've got false and false for the first two because they start on positive numbers and then they're adding a positive number.

So you're never gonna get down into the negative numbers.

C and D are both true though because they're both decreasing sequences.

So again, if we decrease enough times, we will get to a negative value.

The common difference is positive.

So the common difference is positive if it's increasing.

So true for the first two, but false for the second two.

Has one as a term.

So definitely true for the first one, it's our first term.

So the second one is false, we start on seven and then we're adding a positive number.

So we're never gonna get back down to one.

C, false, we can see that we've got the multiples of five, haven't we? So we won't have one in that sequence.

And then D, that's actually true, we're subtracting three each time and if we do that enough times, we do get to the term one.

25 is a term.

So the first one, yes, true.

Well done if you counted through your terms until you got to 25.

B, true, yes, we can see it in our sequence.

C, true, we've already talked about how that has all the numbers in the five times table from 80 downwards.

And then false, we've got our sequence starting on 22 and then decreasing so we can't have 25 as a term.

Bottom row has values greater than 50.

True, because we are increasing for A and B, so they're both true.

C, yes, we can see some of those terms are greater than 50.

And D, false, 'cause we're starting on 22 and decreasing, so we can't have a term that's greater than 50.

Lots of good thinking there guys, well done.

Let's bring that all together then and look at what we've talked about today.

So an arithmetic or a linear sequence can be uniquely described using an initial term and a common difference.

We can see if a sequence is arithmetic by calculating the differences between successive terms. And if we know the common difference and we know one of the terms, we can use that to calculate an initial term.

Fantastic lesson and I look forward to working with you again.