# Lesson video

In progress...

Hi everyone, and welcome to today's lesson on a growing pattern sequences, following on from the rest of our sequences lessons that we've done so far.

Before we begin, once again make sure you've got pen and paper.

As I said, it's really important that you write things down as we go, to help you work things out because no one can do all this in their head, not even me.

So make sure you've got pen and some paper, as well as getting rid of any distractions and trying to find a quiet space if you can.

Pause the video to make sure you've got all of this ready.

No, you're ready to begin, let's start.

We've got Yasmin and Zaki here who were discussing the number of squares in each term of the growing pattern.

Yasmin is saying that this isn't an arithmetic sequence.

We can't work out what the nth term will be.

Zaki is saying, I think I know how many squares the fifth, sixth, et cetera term will have, so we can work out any term.

What I want to know is who you agree with and why? You might want to look back at old videos or in your notes to check what arithmetic means.

Pause the video to do that.

So actually there were a little bits of both.

Yasmin is mean it's correct that this is not an arithmetic sequence.

Remember that means having a constant difference between each term, because we can see that we've got one, we've got four, we've got nine, and 16, and already we can see the difference here is three, five, and then seven.

Zaki is correct that he can find any term because actually you may have noticed and really well done if you have.

These are square numbers, and that's why they actually form the shape of a square.

So we could do one squared to find this, two squared to get four, three squared to get nine, and four squared to get to 16.

So we could find any term using n squared.

100 squared would get you the hundredth term for example.

Some growing patterns are arithmetic and some aren't.

Which of these growing patterns below are arithmetic? And what are the nth term rules for the number of squares? Now the first two, I'm expecting everybody to be able to find the nth term of these two sequences.

The third one extra special, well done.

Think, but maybe to the try this, to find the nth term for that one.

Pause the video to have a go at that.

So the first two are arithmetic, which is why it is a little bit easier to find the nth term.

Each one is going up in four, so I know that I've got four n, and then it's actually four n subtract three.

That's a mistake.

Four n subtract three will get you the first term, four n three will get you the second term because it's the nth term.

Well done if you spotted that.

For the second one, it is also arithmetic and we are adding two each time to get from term to term.

And our nth term is two an add one.

Well done if you got that.

And the last one is not arithmetic, we don't have a constant difference.

But it's actually really linked to the try this task, because you might have noticed in the middle of these patterns, you've got the square numbers.

And each time you go to a new term, we go through the squared number.

So the terms squared, and we also have four extra.

And that four extra stays there.

So actually our nth term was n squared out of four.

So really amazing, well done if you manage to get that one too.

First question is to find the nth term of this sequence here with the growing patterns.

In order to do that, we could see hopefully that we added two each time.

So we should be getting two n and then to get from two n to this, this sequence here, we have to subtract one.

You also draw shapes for the second question to illustrate the first four terms of this sequence.

Three n subtract two.

For this one, there were quite a few different ones you can do and we're going to see an example.

So this is an example that I came up with.

There's lots of different shapes we could have had.

The third question you were asked to find out, which is greater out of two different terms of two different sequences.

The eighth term of four n subtract three, or the fifth term of five n subtract two.

In order to do that, hopefully you recognise that to find the eighth term, that means the position or n is going to be eight, so we've got four multiplied by eight subtract three.

For the fifth time, we've got five multiply it by five, subtract two.

And the eighth term here was bigger as it was 29.

And the rest of the answers are there.

Really well done if you manage to get those correct, especially if you managed to get this 100th term, because it's not one that you could just carry on and workout, you have to use the n to help you down the nth term.

So now we move on to the explore task.

How could you count the total coloured squares in the growing pattern? How many squares will be in the next term and how many squares will be coloured in the 10th term? So here coloured squares just mean the blue ones and the pink ones, just so you don't include those white ones in there.

And look at the way they're coloured because that's a little bit of a hint as to what you can use to help you.

You may have guessed by now looking at the rest of the lesson, it probably isn't going to be your standard arithmetic sequence.

So maybe you think outside the box a little bit.

Pause the video now, let's have a go at that.

So hopefully you managed to see the pattern here.

We can see we've got one here, five here, two, four, six, eight, 10, 12, 13 here, and we might see there's a difference of four here, and then there's a difference of eight here.

And in this one, we've got two, four, six, eight, 10, 12, 14, 16, 18, 20, 22, 24, 25.

There's a difference of 12 here.

So actually you might have started to see a pattern in what is going up here.

But actually these coloured squares produce an even more interesting idea, how to find that pattern.

You can see we've got one pink square and four blue squares here.

You can see we got four pink squares here.

And one, two, three, four, five, six, seven, eight, nine blue squares here.

What can we start to see about these numbers? Where have we seen these numbers before? Here we've got nine pink squares and we've got 16 blue squares.

Hopefully you've noticed that those are the square numbers.

So actually to find any, any pattern in the sequence any term, we just do the position squared.

So for the second one we do two squared, and we'd add the previous term squared.