Lesson video

Hi, Mr. Robson here.

Welcome to Maths.

What a lovely place it is.

Today, we're checking and securing our understanding of non-linear sequences.

I love sequences and I can't wait to see which non-linear ones we're going to examine.

So let's get started.

Our learning outcome is I'll be able to appreciate that not all sequences are linear.

Some keywords for today's lesson that is important, you're familiar with, arithmetic or linear.

And arithmetic or linear sequence is a sequence whereby the difference between successive terms is a constant.

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

This is an arithmetic sequence with a common difference, +2.

Whereas, 10, 12, 15, 19, 24, this is not an arithmetic sequence because there's not a common difference between successive terms. Other sequences we'll see, geometric.

A geometric sequence is a sequence with a constant multiplicative relationship between successive terms. Two parts to today's lesson.

And we're going to begin by looking at those sequences, ones we call arithmetic and geometric.

Let's start with something really familiar.

What do you notice about these sequences? Pause, make a comment to the person next to you, or make some notes to yourself.

See you in a moment.

Welcome back.

Lots you could have said about these sequences.

You may have pointed out that they are the start of the 3, 5, and 9 times tables.

You may then have gone on to elaborate that they all have a common additive term-to-term difference.

For example, your 3 times table has a common additive term-to-term difference of +3.

Your fives, a common additive term-to-term difference of +5.

And no surprise to see your nines have a common additive term-to-term difference of +9.

What if we altered these terms in some way? Taking the first five terms of the sequence 3, 6, 9, 12, 15 and adding 10 to each of them, we'd get 13, 16, 19, 22, 25.

Let's do something similar with that 5 times table, that minus 4 from each term, and we'd get 1, 6, 11, 16, 21.

What about if we multiplied each term in our 9 times table by 2? We'd get 18 36, 54 72, 90.

What do you notice about these sequences that we've generated? Pause, tell the person next to you, or make some notes to yourself.

Welcome back.

I wonder what you said about these sequences.

Did you notice that despite adding 10 to each term, we still had a common additive term-to-term difference of +3 for that top sequence? We minus 4 from each term in our 5 times table, but we still had a common additive term-to-term difference of +5.

When we multiplied each term in the 9 times table by 2, we changed the common additive difference from 9 to 18.

But, importantly, they all still have a common additive term-to-term difference.

Sequences with a common additive term-to-term difference between terms are called arithmetic sequences.

You'll also see them called linear sequences.

Lucas and Aisha are discussing these three sequences.

Lucas says, "From what I've seen so far, I think arithmetic sequences only contain whole numbers, so these sequences cannot be arithmetic." Hmm.

Do you agree with Lucas? Pause this video.

Examine those sequences.

I'll see you in a moment.

Welcome back.

I wonder what you said.

Did you agree with Lucas? Did you disagree? Let's have a look.

Aisha has a useful argument.

She says, "Let's look at the definition of an arithmetic sequence again, Lucas." That definition being, sequences with a common additive difference between terms are called arithmetic sequences And when we examine these three sequences, we find the first sequence has a common additive difference of negative 4.

Our second sequence was all negative terms, but a common additive difference of +3.

And our third sequence was fractions, but it had a common additive difference of +3/7.

So Lucas concludes, "I was wrong! These are arithmetic sequences.

The common difference can be a fraction.

It can be negative." Let's check you've got that.

Four sequences here.

Which of these are arithmetic? Pause this video and see if you can work that out.

See you shortly.

Welcome back.

Let's see how we did.

We're looking to identify which of these sequences are arithmetic.

The first one, absolutely.

A is arithmetic.

There's a common difference of +6 between successive terms. B was tricky, decimals, but it's arithmetic.

We have a common difference of +0.

05 between successive terms. C was made up of mixed numbers, but it's arithmetic.

Common difference of positive +1/2 between successive terms. And I wonder what you said about D.

Look at those denominators, 2, 4, 6, 8, 10.

That's our 2 times table.

So is this arithmetic? No.

It may look a little bit arithmetic when we first glance at it, but on further study, it's not arithmetic.

There's no common difference.

If you look at the sequence, 1/2, 1/4, 1/6, et cetera, you'll see differences, term-to-term differences, of -1/4, -1/12, -1/24, -1/40.

That's not not a common difference, so we can't call that sequence arithmetic.

Lucas and Aisha are looking at a different kind of sequence.

1, 2, 4, 8, 16.

"This sequence is different," says Aisha.

"The difference between terms is the sequence itself!" Hmm.

I wonder what she means by that.

When we examine this sequence and look at the term-to-term additive differences, it goes add 1, add 2, add 4, add 8, and the sequence itself was 1, 2, 4, 8, et cetera.

Hmm.

How interesting.

Lucas has spotted something.

"I've seen these sequences before.

1 x 2 = 4, 2 x 2 = 4, 4 x 2 = 8, 8 x 2 = 16.

The term-to-term rule is multiplied by 2." Aisha says, "That's right.

A common ratio of 2 between terms. This sequence is geometric!" With a geometric sequence, it's best to think of that term-to-term difference as multiplicative rather than additive.

Geometric sequences have a constant multiplicative relationship between successive terms. Lucas and Aisha are looking at another different kind of sequence.

2, 6, 18, 54, 162 are the first five terms. Lucas says, "Can you spot the multiplicative relationship between terms here?" Aisha says, "Yes, it's times 3." There you can see that times three between successive terms. 2 times 3 makes 6, 6 times 3 makes 18, et cetera.

Lucas says, "We call this multiplier the common ratio." In geometric sequences, the constant multiplicative relationship between successive terms is called the common ratio.

We'd say that sequence, 2, 6 18, 54, 162, are the common ratio of 3.

Lucas then says, "This one is not so easy.

How will we spot the common ratio here?" The sequence going 3.

26, 13.

04, 52.

16, et cetera.

But Aisha says, "Easy! Divide each term by the previous term." You might use your calculators to do this, but we take the fifth term, 834.

56, divided by the fourth term, 208.

64, and we get 4.

And as we do the same for the fourth term divided by the third term, the third term divided by the second term, the second term divided by the first term, we get 4 every time.

Lucas says, "Excellent work, Aisha! A common ratio of 4." Well done the two of you.

Quick check you've got that.

Divide each term by the previous term to find the common ratio of this sequence.

The sequence starting 1,237, 7,422, 44,542, and so on.

Pause this video.

See if you can find the common ratio for this sequence.

See you in a moment.

Welcome back.

Let's see how we did.

Did we start with the fifth term divided by the fourth term, then the fourth term divided by the third term, the third term divided by the second term, the second term divided by the third term, and did we get six every time? I hope so.

We would say its geometric sequence has a common ratio of 6.

Another one for you, and a slight difference this time.

I wonder if you'll spot what's going on.

The sequence going 768, 567, 432, et cetera.

Again, I'd like to divide each term by the previous term to find the common ratio.

Pause and do that now.

Welcome back.

We should have started with the fifth term divided by the fourth term, fourth term divided by the third term, and so on, and we should have got 3/4 each time.

We would say this is a geometric sequence with a common ratio of 3/4.

Beware, that common ratio will not always be a whole number nor an integer.

We might see fractional common ratios, and they give us beautiful sequences like this one.

Practise time now.

For question one, I'd like you to categorise these eight sequences into arithmetic and geometric.

Pause and do that now.

For question two, I'd like you to find the blanked out first and fourth terms of these sequences.

Now, pay careful attention to the language here, because at first glance, a and b look like the same question.

They are not.

In a, it's specified that these 9 and 27 are part of an arithmetic sequence, and in b, it's specified that 9 and 27 are part of a geometric sequence.

Pay particular attention to that vocabulary.

Pause.

I'd give this problem a go now.

Welcome back.

Let's see how we did.

Question one, I asked you to categorise some sequences into arithmetic and geometric.

We should have put a in the arithmetic box 'cause it has a common difference of +2 between successive terms. B should have gone in the geometric box, has a common multiplicative relationship between terms, we'd say it's got a common ratio of 2.

C is in the arithmetic box, has a common difference of +8 between successive terms. D is geometric, a common ratio of 3.

E had common additive difference of -12 between terms. It's arithmetic.

F, slightly trickier one, but a common ratio of 1/3 between successive terms. That makes it geometric.

G, an unusual sequence, but a beautiful one.

It's a common ratio of -2.

1 multiplied by -2 makes -2, -2 multiplied by -2 makes +4, and the sequence oscillates.

As I'm going to identify, it's a common ratio of -2.

That makes it geometric.

Finally, h.

A common difference between successive terms of -3 makes it arithmetic.

For question two, Find the blanked first and fourth terms of these sequences.

We're told that sequence a is arithmetic.

Between 9 and 27.

we'd need a common additive difference of +18, and if that additive difference is repeated throughout that sequence, then the fourth term must be 45 and the first term must have been -9.

For part b, we're told it's a geometric sequence.

The multiplicative relationship between 9 and 27 is to multiply by 3, so we've got a geometric sequence with a common ratio of 3.

That would make the fourth term 81, and the term before 9 must have been 3.

For part c, it's an arithmetic sequence with a common additive difference of -3.

That makes the fourth term -2, and the first term must have been +7.

But if we had the terms 4 and 1 in a geometric sequence, we'd have a common ratio of 1/4.

That'll make our fourth term 1/4 and the first term 16.

Onto the second part of our lesson now, where we're going to look at other types of sequences.

I wonder what they're going to look like.

Let's take a look.

Aisha and Jacob are studying another sequence.

1, 4, 9, 16 and it continues.

Aisha says, "I'm stuck finding the fifth term here.

This sequence has no common difference and no common ratio." Lucas says, "I can help! This pattern is our square numbers!" 1 squared is 1, 2 squared is 4, 3 squared is 9, 4 squared is 16.

Aisha says, "So the next one is 5 squared.

That's 25.

Thanks, Lucas!" Not all sequences have a common difference or common ratio.

There are other sequences beside arithmetic and geometric ones.

Aisha takes a closer look at this sequence, our square numbers.

And she says, "This is interesting.

The term-to-term differences have a common difference!" I wonder what she means.

When you look at the term-to-term additive difference between our square numbers, add 3, add 5, add 7, add 9, oh, there's a pattern there.

We have a common second difference of +2.

Unlike arithmetic sequences, which have a common first difference, some sequences, like our square numbers, have a common second difference.

Triangular numbers enjoy this same feature.

Square numbers go 1, 4, 9, 16, 25, and so on.

Our triangular numbers, 1, 3, 6, 10, 15, and so on.

We look at the first difference, and it goes add 2, add 3, add 4, add 5.

Aha! A common second difference.

A common second difference of +1 for our triangular numbers.

Aisha and Lucas experiment with these special sequences, our square numbers and our triangular numbers.

Aisha says, "I'm going to multiply the square numbers by -2." 1 multiplied by -2, 4 multiplied by -2, 9 multiplied by -2, would generate that sequence.

Lucas says, "I'm going to add -5 to the triangular numbers." 1 add -5 makes -4, 3 add -5 makes -2, et cetera, to get that sequence.

What do you notice about the sequences that Aisha and Lucas have created? Tricky question, but you'll get there.

Pause this video, have a little play with those sequences, see what you notice.

See you in a moment.

Welcome back.

How did we get along? What did we notice about these sequences? Did you look at the additive difference between successive terms and get something like this? Aisha and Lucas did, and they both concluded, "We still get a common second difference." How beautiful.

Quick check you've got that now.

Three sequences there.

Some of them have a common second difference.

Some of them do not.

Which of these sequences have a common second difference? Pause this video.

Work that out now.

Welcome back.

Let's see how we did.

A does indeed have a common second difference.

There it is, +20.

B does not.

There's a common difference, but it's not the second difference.

It's a common first difference.

C does.

A common second difference of +3.

Aisha and Lucas examine another special sequence.

1, 8, 27, 64, 125.

Do you recognise those numbers? Well done.

They are indeed your cube numbers.

Lucas says, "If the square and triangular numbers have a common second difference, I wonder what we'll find for the cube numbers." Let's look at the additive difference between terms. And I don't see anything there, do you? Let's look further.

There's no common second difference.

Have you spotted it? Well done A common third difference in our cube numbers.

Aisha says, "Wow! A common third difference! That is cool." And I couldn't agree more, Aisha.

Our cube numbers and related sequences to our cube numbers have a common third difference.

Quick check you've got that now.

Find the common third difference of this sequence.

5, 8, 18, 40, 79, and so on.

Pause.

Find that common third difference.

See you in a moment.

Welcome back.

How did we do? Did we find the first difference, the second difference, and the third difference, and find that it is indeed a common third difference of +5? Practise time now.

I'd like you to identify which of these sequences have a common second difference and which of them have a common third difference? Pause and do that now.

Question two.

Given that these sequences have a common second difference, find the missing terms. Pause and do this now.

For question three, Jacob is examining these three sequences, sequence a, sequence b, sequence c, and Jacob says, "I see three increasing sequences whereby the terms will always go up." Jacob's statement is not entirely true.

Explore the sequences and see if you can find the contradiction.

Pause and do that now.

Welcome back.

Feedback time now.

For question one, I ask you to identify which of these sequences have common second difference and which have a common third difference.

We should have noticed for a, there was a common second difference of +9.

We should have noticed for b, not a common second difference, but a common third difference of +1.

For part c, we should have got those first differences, those second differences, and spotted a common third difference of -8.

And for d, that's the first differences leading us to a common second difference of +10.

For question two, these sequences have a common second difference.

Knowing that is the key to unlocking those missing terms. Same, we should have noticed a common second difference of +2 for part a.

That would mean the first differences go at 4, at 6, at 8, at 10, giving us missing terms of 19 and 29.

For part b, we should have noticed the differences giving us a common second difference of +7.

If we do 23 minus 9, we get back to 14, 14 minus 2, we get back to 12, and your sequence looks like that.

For part c, a common second difference of -4, giving us a fifth term of 56 and a first term of 0.

And part d was a lovely sequence.

It goes add 1/4, add 3/4.

If it's got a common second difference, that second difference must be 1/2, so your sequence would increase like so.

The fourth term, you might have written 10/4.

We would cancel it down to 5/2.

If you've written 10/4, that's mathematically fine, but we'd always simplify that answer, and the next term being 17/4.

Question three, and a lovely question indeed.

Because at first glance, Jacob's right, three increasing sequences.

The terms are going up.

It's hard to disagree, unless we further examine.

The first sequence is arithmetic, with a common first difference of +8 between terms. They absolutely will increase forever and ever in that sequence.

There's a few terms, and we can see from that, it will always increase.

For sequence b, there's no common first difference, but there is a common second difference of +2.

Positive second difference forever increasing the first difference, so the terms will increase forever.

I can write out the next few terms in the sequence, and you can see that first difference growing and growing and growing.

Those terms are only going to get more and more positive and larger and larger.

C was different.

Can you see the difference? A common second difference of -2.

So how will this one be different? Negative second difference is decreasing the first difference.

If we run that sequence on, we'll get to the point where the term-to-term additive difference is 0.

This sequence will reach that.

We've got successive terms of 112 and 112.

It's no longer increasing.

What would happen into the future? We keep that common second difference and -2 going, all of a sudden our first difference becomes negative and this sequence starts to decrease.

Jacob says, "Wow! A sequence which was increasing and then starts decreasing! That is cool." Jacob, you're right.

That is cool.

Sadly, we're at the end of the lesson now.

In summary, sequences with a common additive term-to-term difference are arithmetic, also known as linear.

Sequences with a common multiplicative relationship from term-to-term are geometric.

We can appreciate that not all sequences are arithmetic or geometric.

There are other types of sequences with other features.

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

Goodbye for now.

(object rustling).