Lesson video

Hello, Mr. Robson here.

Welcome to Maths.

Great choice to be here, checking our understanding of arithmetic sequences.

Sequences, yay! Who doesn't love sequences? No one.

So let's get learning.

Our learning outcome is I will be able to appreciate that any term in an arithmetic sequence can be expressed in terms of its position in the sequence, and we'd be able to generate any term.

Keywords that you'll hear me using throughout this lesson, arithmetic sequence, linear sequence.

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

For example, the sequence 10, 12, 14, 16, 18, has a constant common difference of +2.

It's an arithmetic sequence with a common difference of +2.

Whereas 10, 12, 15, 19, 24, this is not an arithmetic sequence because there is not a common difference between successive terms. In addition to that, you'll hear me using the phrase nth term.

The nth term of a sequence is the position of a term in a sequence where n stands for the term number.

For example, if n = 10, this means the 10th term in the sequence.

Look out for that language throughout.

Three parts to today's lesson to check your understanding of arithmetic sequences.

We'll start with identifying them.

Sofia is looking at some number sequences.

These sequences.

She says, "Whilst these are all different sequences, I can see something they all have in common." Can you see what Sofia has noticed? Pause this video and have a look at those three sequences.

Welcome back.

Did you spot what Sofia had spotted? A common difference of +5 in the top sequence, a common difference of +4 in the second sequence, and a common difference of +7 in the third sequence.

"All three sequences increase in equal steps each time." Sequences with a common difference are called arithmetic sequences.

These are three examples of arithmetic sequences.

You may also hear them called linear sequences.

Sofia is testing Lucas on arithmetic sequences.

She uses these four sequences.

She says, "One of these sequences is not arithmetic.

Which one do you think it is?" Lucas says, "I think it's c because that one is decreasing." Do you agree with Lucas? Pause this video.

Have a conversation with the person next to you or a think to yourself.

Do you agree with Lucas's statement? Welcome back.

Sofia says, "Let's look at the term to term differences, Lucas." In sequence a, common difference of +7.

In sequence b, common difference of +20.

In sequence c, a common difference of -5, but it's still a common difference, whereas sequence d, +5, +10, +5, +10.

It's a lovely pattern, but it's not a common difference.

Lucas says, "I think I've changed my mind.

Is it d that is not arithmetic?" "Yes, Lucas, that's right," says Sofia.

Sequences a, b and c are arithmetic.

They all have a common difference.

It doesn't matter that c has a negative common difference.

It's still an arithmetic sequence, whereas d is not arithmetic.

The differences are not constant.

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

Which of these are arithmetic sequences? Three for you to ponder, which are arithmetic, which are not? Pause and think about that now.

Welcome back.

Did you spot a, is arithmetic.

It's an arithmetic sequence with a common difference, -12 between successive terms. B was also arithmetic.

It's an arithmetic sequence with a common difference of +4 between successive terms. C was not.

There is no common difference between those terms. It's not arithmetic.

Practise time now.

I'd like you to label these sequences as arithmetic or not arithmetic.

That's all you need to do.

Eight sequences.

Some of them are arithmetic, in which case, write arithmetic next to them.

Some of them are not.

You might want to add a little detail to justify why the ones that are not arithmetic are not arithmetic.

I'll leave you to that now.

Pause and give these a go.

Feedback time.

Let's see how we did.

For a, it was arithmetic, a common difference of +4.

B, it was also arithmetic despite a decreasing sequence, a common difference of -6, it's still arithmetic.

C was also arithmetic, all negative terms, but it doesn't matter.

It's arithmetic.

It's got a common difference.

+2.

We were due one that wasn't arithmetic.

D is not arithmetic.

No common difference, not arithmetic.

E's an unusual one in that they're not integer values.

Does that matter? It's a common difference of +1.

1 between successive terms. That's arithmetic.

For f, it's a lovely sequence, but it's not arithmetic.

G and h were unusual, fractions.

Can we have that in arithmetic sequences? Of course we can.

+5/29 is the common difference in g, making it an arithmetic sequence.

I wonder if a few people were caught out by h, the numerator is all one, the denominators, 2, 3, 4, 5.

2, 3, 4, 5, would be an arithmetic sequence, but as the denominators of fractions, no, because the difference between a 1/2 and a 1/3 is -1/6.

The difference between a 1/3 and a 1/4, -1/12, it's not a common difference.

That's not an arithmetic sequence.

Moving on to part two of the lesson, generating arithmetic sequences.

Arithmetic sequences can be expressed in terms of the relationship between the term number, which we call n, and the term value, which we can call T.

If I put them into a table of values, I've got the first five terms of a sequence that goes 3, 6, 9, 12, 15.

In the case of this sequence, when n=1.

T=3, you can read that from the table.

In the n position of one, look down from there, you'll see a T value of three, so what do I mean when I say when n = 1, T = 3? It means the term in the first position of the sequence has a value of three.

When n = 2, T = 6, the term in the second position has a value of 6.

We can keep going with this.

When n = 3, T = 9.

The term in the third position as a value of 9.

I'd like you to keep going with that.

I just wanna check you've got that.

It's the same sequence.

I'd like to fill in the blanks at the bottom.

When n = 5, T = what? And this means the term in the which position has a value of what? Pause.

Fill in those blanks.

I'll see you in a moment.

Welcome back.

I hope you said when n = 5, T = 15 We can read that from the table.

This means the term in the fifth position has a value of 15, well done.

In the case of the sequence 3, 6, 9, 12, 15, there's an obvious relationship between the term number and the term value.

That obvious relationship being multiply by three, take the term number n = 1, multiply that by three, a term of three, the term value T is always three times the term number.

This is written more simply as an nth term of 3n.

If we were able to simplify that last sequence to an nth term 3n, what can we do with this one? I'd like you to fill in those blanks.

I'd like you to identify the multiplicative relationship between the term number n and the term value T for this sequence.

I'd like you to fill in the blanks on the last two sentences on this slide.

Pause and have a think about that now.

Welcome back.

Did you notice the multiplicative relationship was times five? We take our term number, multiply it by five to get our term value so we could say the term value T is always five times the term number.

This is written more simply as an nth term of 5n.

In this case, having an nth term of 5n enables us to find any term in the sequence by substitution.

I've already given you the first five terms, 5, 10, 15, 20, 25, but what if we wanted to know terms in the future of this sequence? We'll use substitution.

We know it's nth term, 5n, i.

e.

, the term value is five lots of the term number.

So we can substitute in any term number.

If we wanted to know the 10th term in the sequence we'd substitute in n = 10.

5 lots of 10 is 50, so that's a term value, 50.

If we want to know the 50th term in the sequence, we substitute n = 50.

5 lots of 50, 250.

That's the 50th term in the sequence, and we can do the same for the hundredth term.

Substitute a hundred into 5n.

5 lots of 100 is 500 The hundredth term in our sequence has a value of 500.

Sometimes the relationship, the nth term, might be two steps.

What if it wasn't just multiply by three or multiply by five? What if it was multiply by four and add one? This means that the term value is four times the term number plus another one.

We just use substitution again.

If the nth term is 4n + 1, the first term is 4 lots of 1 + 1, that's 5.

The second term is 4 lots of 2 + 1, that's 9.

4 lots of 3 + 1, 13.

4 lots of 4 + 1, 17.

4 lots of 5 + 1, 21.

From an nth term, we've generated the first five terms. Let's check you've got that now.

I'd like you to find the first five terms and then the 50th term of the sequence 2n + 5.

That means take your term number, multiply by two, add five to get your term value.

You might want to use the modelling I've left at the bottom of the screen there for the first two terms. I'd like you to pause now and give this a go.

Welcome back.

Let's see how we did.

The nth term, 2n + 5.

Well, we'll substitute in n = 1 for the first term.

2 lots of 1 + 5.

That's seven.

For the second term, 2 lots of 2 + 5, that's 9.

For the third term, 2 lots of 3 + 5, 2 lots of 4 + 5, 2 lots of 5 + 5, 2 lots of 50 + 5.

We get the first five terms, 7, 9, 11, 13, 15, and a 50th term of 105.

Practise time now.

I'd like you to find the first five terms of these four sequences.

The sequences 3n + 7, 3n - 7, 0.

3n + 7, and 7 - 3n.

Use substitution.

Find the first five terms of those sequences.

Pause now and do that.

For question two, I'd like to use substitution to find the 10th, the 50th, and the 100th terms of these sequences.

You might choose to use your calculator for some of these.

For example, if you were using your calculator on part c, 0.

9n + 0.

6, when you're substituting in 10, you can just put 10 in a bracket like that.

Your calculator will do 0.

9 times 10 for you.

You could then use the left right arrow keys and the delete button to change that 10 into a 50 for the 50th term, to a hundred for the hundredth term.

Whether you use your calculator is up to you.

I quite like not using my calculator because I like the mental exercise of trying to do these without, and then I use my calculator afterwards to check my results.

The choice would be yours.

Anyway, pause this video and find the 10th, 50th, and 100th terms of those sequences.

See you in a moment.

Feedback time.

The first five terms of these four sequences.

For part a, the sequence 3n + 7 starts 10, 13, 16, 19, 22.

For B, the sequence 3n - 7 starts at -4, and then - 1, 2, 5, 8.

Sequence c, 0.

3n + 7.

That goes 7.

3, 7.

6, 7.

9, 8.

2, 8.

5 and d.

That sequence goes 4, 1, -2, <v ->5,</v> <v ->8.

</v> You might want to pause and just check that your terms match with mine.

For question two.

Part a, the 10th, 50th and hundredth terms of the sequence 9n + 6.

Well, I'm substituting a 10 into 9n + 6, 9 lots of 10 + 6, that's 96, and then I'm substituting in n = 50.

9 lots of 50 + 6, that's 456.

Then I'm substituting in n = 100 for the hundredth term.

9 lots of 100 + 6, that's 906.

For part b, 9n - 6.

I'm substituting in 10 again for the 10th term, 9 lots of 10 - 6, that's 84.

I'm substituting in 50 for the 50th term.

9 lots of 50 - 6 is 444, and then for the hundredth term, 9 lots 100 - 6 that's 894.

You again may want to pause just to check that you've got the same method that I have and the same terms that I have.

For part c when n = 10, the term is 0.

9 X 10 + 0.

6, that's 9.

6.

For the 50th term, 0.

9 lots of 50 + 0.

6, that's 45.

6.

The hundredth term, 0.

9 lots of 100 + 0.

6 is 90.

6.

For part d, the sequence 6 - 9n.

6 - 9 lots of 10 is -84.

For the 50th term, 6 - 9 lots of 50, that's -444.

For the 100th term, 6 - 9 lots of 100, that's -894.

Again, pause, check that your method matches mine and your terms match mine.

Part three of the lesson now.

Representations of arithmetic sequence.

I love representations, so let's get started.

How is this pattern growing? <v ->Hmm,</v> hmm, hmm.

Can you see how the pattern is growing each time? Well done.

It's constantly adding three counters.

The next pattern would be that fourth pattern plus another three counters.

This is an example of a visual representation of an arithmetic sequence.

If we count the counters, the sequence is 2, 5, 8, 11, 14.

You can see the +3, the common difference, in the numbers, and you can see it in the pattern, the visual representation to the addition of three counters each time.

Arranging the same sequence in a different way shows us something about the underlying structure of arithmetic sequences.

If I start with the two counters that we saw, and then I show you the five counters slightly differently, can you remember what's coming next? Well done.

Eight counters and then 11 counters.

You can see the constant additive difference more easily, the addition of three each time, and you can also see why they're known as linear sequences.

Can you see why they're called linear sequences? I hope so.

We'd see the same linear structure if we plotted the sequence graphically.

That's a table of values for the relationship between Term number and Term, and we can plot these by labelling our axes Term value T and Term Number n.

That's a graph of this linear sequence.

The horizontal axis, Term number n, the vertical axis, Term value T, and there're the coordinates.

1, 2, 2, 5, 3, 8, 4, 11 and 5, 14 as red from the table.

Can you see why they're called linear sequences? Whilst we can see that the points align, we only plot the coordinates.

We don't draw a straight line here, but it's a nice way to check for the linear sequence.

Let's check you've got that.

This sequence will plot points in a straight line, 10, 16, 22, 28, 34.

True or false, that'll plot points in a straight line.

When you've decided whether it's true or false, I'd like you to justify your answer with one of those two statements at the bottom.

I'd like you to pause, have a think about that now.

I hope you said true and justified that with a, 10 16, 22, 28, 34 is an arithmetic sequence.

It's got a common difference of +6 between successive terms, therefore it's linear.

The points will align.

Andeep and Laura are trying to plot the arithmetic sequence starting 34, 28, 22, 16, 10.

Andeep says "This sequence will have negative terms" Sequence will have negative terms? Oh of course, if we continue that common difference of -6, eventually this sequence will run into the negatives.

so Andeep says, "The sequence will have negative terms so it can't be plotted." What do you think? Do you agree? I'd like to pause this video, have a conversation with a person next to you or a think to yourself.

Do you agree with Andeep's statement? See you in a moment.

Laura, partially agrees.

"You are right.

There will be negative terms." The sequence continues 4, -2, -8, -14, "But we can plot those in the fourth quadrant." What does Laura mean by that? Instead of just plotting in the first quadrant where the term numbers are positive and the term values are positive, we can use the fourth quadrant of our coordinate grid and that enables us to plot negative term values.

Andeep says, "It can be plotted.

We just use the fourth quadrant." as I've highlighted there.

"Thanks, Laura." Well done you two.

Let's check you've got that.

True or false? The negative terms of the sequence, <v ->13,</v> <v ->8,</v> <v ->3,</v> 2, 7, cannot be plotted.

Is that true or false? Once you've decided whether it's true or false, I'd like you to use one of those two statements at the bottom to justify your answer.

Pause this video, do that now.

I'll see you in a moment.

Welcome back.

I hope you said false.

The negative terms of the sequence can be plotted.

How do we justify the fact that's false? We use statement b, they can be plotted using the fourth quadrant, the first term of -13, the second term of -8, the third term of -3, we plot in the fourth quadrant, like so.

Practise time now.

Question one.

I'd like you to draw a visual representation for the first five terms of these sequences and by visual representation, I don't mean draw a graph, I'd like a visual representation, maybe like the one I used counters for earlier.

Draw visual representation, something like those counters, for each of these sequences.

Pause, get drawing.

I'll see you in a moment.

For question two, I'd like you to plot the first five terms of these sequences.

The sequence 4n - 3, and the the sequence 10 - 3n.

You will first have to use the table of values to generate the first five terms before you can plot them on the respective graphs.

Pause and give this a go now.

Feedback time.

These are just examples of what they might look like.

I see 5, 7, 9, 11, 13.

Changing like this, there's five counters in the first column.

Seven counters in the second column, nine counters in the third column, 11 counters in the fourth column.

13 counters in the fifth column revealing the linear structure of that sequence.

I drew something similar for the sequence 13, 10, 7, 4, 1, start with 13, then 10, then seven, then four, then one.

That's quite nice.

It reveals the linear structure as well, and also the fact it's a decreasing sequence.

For part c.

I changed my visual representation a little bit.

That's the sequence.

6, 11, 16, 21, 26, and I see the first term like that, a one with five on top.

The second term I see as a one with two lots of five.

The third term a one with three lots of five, a one with four lots of five, and a one with five lots of five.

Again, revealing that linear structure to that sequence.

For question two, the first five terms of the sequence, 4n - 3.

Those terms are 1, 5, 9, 13, 17, so we plot the coordinates 1, 1, 2, 5, 3, 9, 4, 13, and 5, 17, like so.

For part b, that sequence goes 7, 4, 1, -2, <v ->5,</v> so we plot the coordinates 1, 7, 2, 4, 3, 1, 4, -2, and 5, -5.

Sadly, that's the end of the lesson now.

I've enjoyed it.

What have we learned? We've learned that any term in an arithmetic sequence can be expressed in terms of its position in the sequence, and we can generate any term using the nth term.

For example, in the sequence 7n - 5 the first term is 7 lots of 1 - 5 which is two, whereas the 10th term is 7 lots of 10 - 5, which is 65.

I hope you've enjoyed this lesson as much as I have, and I hope to see you again soon for more mathematics.

Bye for now.