Lesson video

Hi, I'm Mr Chan, and in this lesson we're going to learn how to write the nth term of a linear sequence.

So we're learning about linear sequences in this lesson.

It's important to understand which sequences are linear, and which sequences are not, so let's have a look at that.

We've got two sequences here, 2, 5, 8, 11, 14 and second sequence 2, 5, 9, 14, 20.

Let's look at what's happening between these terms. So to get from 2 to 5, we're adding 3.

Then we're adding 3 again to get from 5 to 8.

8 to 11 we would add 3, and finally add 3 again.

We can see that this sequence is going up by the same amount each time.

The second sequence, 2 to 5 we're adding 3.

Then from 5 to 9 we're adding 4.

Then from 9 to 14 we're adding 5.

And then adding 6 to get from 14 to 20.

This sequence is not going up by the same amount each time.

It's not increasing by the same amounts.

So what we can say is that that sequence is not linear, because a linear sequence is a sequence that will increase or even decrease by the same amount each time.

Here's an example of how we find the nth term for a linear sequence, and let's begin with this sequence we've got in front of us.

2, 5, 8, 11, 14.

Now we refer to the sequence numbers with a position number, and that's what we call "n".

So in position one, we have the term 2, in position two we have 5, in position three we have 8, etc.

So, we can see with this sequence it's increasing by 3 every time.

Now, the only other thing that we know in maths that increases by 3 each time is the 3 times table.

So what we have to do with that then, is say the sequence will begin with 3 times table, and we get that from multiplying 3 times n.

So we write 3n.

Now, the sequence isn't quite the 3 times table, because we can see otherwise, it would just match up with the 3 times table.

It differs very slightly from the 3 times table.

As we can see the 3 times table will start with 3, whereas the sequence starts with 2.

And then it goes on to 6, the sequence is 5.

What you notice is that how it differs from the 3 times table is it's actually 1 less than 3 times table.

So 3 take away 1 will get you to 2, 6 take away 1 will get you to 5, etc.

It's always 1 less.

So how we write that sequence as a nth rule, nth term, is this sequence will follow the 3 times table, but it's always 1 less, so we write 3n subtract 1.

Here are some questions for you to try.

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Here are the answers.

In question one, just a reminder what linear sequence is, a sequence is linear if it increases or decreases by the same amount each time.

In question two you're asked what is the nth term for each sequence, and you'll notice that they're all linear and what's happening in each one of those questions is that they're increasing by 4 each time, so each one of those sequences begins with 4n.

Now, they all differ slightly with the 4 times table, in fact, part A matches with the 4 times table, so we actually just write 4n for that sequence.

In parts B and C they differ slightly with the 4 times table, in part B it's 3 more than the 4 times table, so we write 4n add 3, and in part C, it's actually 7 less than the 4 times table that sequence, so we would write for the nth term, 4n subtract 7.

Here's a question for you to try.

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Here's the answer for question three.

When you're presented with picture patterns, a really good idea is to try and link the pictures into numbers, and that's what that table there is trying to do to help you.

So it's looked at the pattern numbers, and it's asking you how many shaded squares there are, and how many unshaded squares there are.

When you can see the sequences, and once you can see that they're linear, you can then create the nth term rule, and from this question, the link between the shaded squares and the unshaded squares, you can see that the nth term is 3n plus 1.

So let's have a look at another example for finding the nth term.

Again, we're going to give each term a position number, starting with one, we always start with one, then two then three.

So this sequence as we can see when we look at what's happening from term to term, this sequence is decreasing by 2 each time.

So something that decreases by 2, it will follow the negative 2 times table, so we can say that this sequence will begin with negative 2n.

Now, how does this sequence match up with the negative 2 times table? Well, if we were to compare the negative 2 times table with the actual sequence, we can see that the first term compared to the negative 2 times table, we must add 4.

So let's do that for the next term.

Again, negative 4 add zero will get us to zero.

Negative 6 add four will get us to negative 2, etc.

So we can say the nth term for this sequence will follow the negative 2 times table add four.

And we write negative 2n add four.

Here are some more questions for you to try.

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Here are the answers.

We can see in question four that Teddy's actually mistaken, he think that sequences can't be linear if they're decreasing, however linear sequences can increase or decrease as long as from term to term they're increasing or decreasing by the same amount each time.

Let's have a look at another example.

We've got some missing numbers in this sequence that we've got to work out.

I think it's important to know that it does tell us the sequence is linear, so that's quite important because we then know that the sequence is increasing or decreasing by the same amount each time.

Because 5 goes up to 26, we can see that this is a sequence that will be increasing by the same amount each time, so let's figure out what the missing numbers are.

There's actually two missing numbers in between the 5 and the 26, so that means that there's one, two, three divisions there.

And we can figure out what the difference is, 21.

So to get from 5 to 26 we get to 21, but we've got to break that up into three divisions.

All right so, this 5 increasing to 26 is broken up into three steps.

We can figure that difference out by dividing by 3, and that will get us 7, so we can figure out that to get from 5 to 26 in equal steps, it's got to be add 7 each time.

Pretty straight forward to figure out what the missing numbers are.

5 then 12 then 19 then 26 and obviously the last number that's missing must also be add 7, to get 33.

Again, to work out the nth term, let's set up and compare position number with the sequence as so.

And what we need to figure out, well we already know that it's increasing by 7 each time, so we can write 7n, because it follow the 7 times table, and decide how the sequence differs from the 7 times table as such.

It's always 2 less.

So the final nth term, we can write it follows the 7 times table, but always 2 less, so we write 7n subtract 2.

Here are some questions for you to try.

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Here are the answers.

Let's look at question seven.

So, we have some missing terms with that sequence, and as discussed in the example to find the missing numbers, what you can do is look at the interval between 5 and 23.

There's an interval of 18 and that's been split up into three gaps, from 5 to a missing number, to another missing number, to 23.

So that's a total of three gaps.

And the interval of 18 divided by 3, that gives you 6.

So that's telling me that once you've figured out that the gaps are increasing by 6, it's adding 6 each time, so that helps you find the missing numbers, and also links to finding the nth term, because it's increasing by 6 each time.

That's all for this lesson.

Thanks for watching.