Lesson video

Hi everybody, and welcome to our fourth lesson on sequences.

Today we are learning about one of my favourite topics which is all about something called the Nth term.

So we're building on the knowledge that we learned in the last lesson about generalising those sequences.

Before we get into that exciting stuff though, you just need to make sure that you've got pens and paper, once again that you've got no distractions in front of you and hopefully you've got a fairly quiet space to work.

So pause the video here to make sure you've got all of that sorted.

And once you've got all of that sorted, we can begin.

All right.

Let's start.

So the first thing I want you to look at is this grid to the right.

Some columns and rows have been torn off this number grid where you can see those jagged edges.

Imagine this grid has six columns.

So at the moment you can see it only has three columns, I want you to imagine it has six.

What numbers would go in the blue squares? How would this change if the grid had eight columns? What about 100 columns? So you're going to think about, back to what we learnt before, think about, if we've got six columns, what does that mean this number is definitely going to be without even filling in those columns out there? You should hopefully know already how I'm going to work out if its got six columns.

So pause the video now to have a go at that.

So hopefully, for the grid with six columns, you managed to get this.

Well done if you did.

Remember we are going up in sixes if its got six columns.

So to get from negative one, we need to add six to get to five and so on.

If you managed to get row 100, extra well done to you.

We are thinking, if I've got this would be six lots of zero, this would be six lots of one, this would be six lots of two.

so row 100 actually should be six lots of 99 in this column here.

This is the column that I start with most of the time because its got my multiples of six in it.

Notice that if this this was a zero, not a six, so that row one is not six lots of one it's actually six lots of zero, which is why we had to adjust row 100 so if you put 600 here you got a bit caught out, but I understand that's no problem, and hopefully, if you noticed it in this bit, you can maybe pause the video here and go back to your other answers and maybe change some bits.

Carrying on, if it had eight columns, then we're looking at something similar.

So this column here, which had zero, we're then going up in eights.

And the same here would actually be eight multiplied by 99, not by 100.

And finally, for 100 columns, it's over here, but we could see that row 2 would have 100 in it, and actually row 100, again multiplied by 99, is going to have 9,900 in it.

And the other two, you just shifted them one space back and one space forward.

So we just added one or subtracted one to each of those.

Really, really well done if you got those, especially the row 100 and 100 columns cause those ones you couldn't just fill out by adding in the numbers and drawing out the full grid.

You couldn't do it that way.

So you had to use that deduction.

You had to use what we've learned about generalising and tracking calculations, so really well done if you did that.

Previously, we have seen how tracking calculations can help us to generalise and find any number in a number sequence.

So we've seen something like this before when we had nine, 15, 21 as a sequence we know the difference in those numbers was six, so it was six lots of something, and something.

Here we can see that if this was number nine, it was six lots of one, add three, six lots of two, add three, cause it's all being shifted, all those multiples of six being shifted three spaces to the right up.

If we take the number sequence nine, 15, 21, 27 we can use multiples and tracking calculations to find the position-to-term rule.

The position-to-term rule is what we're going to be looking at today.

And it's also known as the Nth term.

And we can see that here: A position-to-term rule can helps us work out any term in an arithmetic sequence.

So we can replace that N with any term, any number, and we'd work out what the number was in that sequence.

So we can see it a bit more formally here.

To find any term in the sequence 9, 15, 21, 27 I can use the position-to-term rule: six n add three And this is being found, because we're looking at multiples of six we're going up in sixes, so it's multiples of six, and then we're just working out where those multiples of six have been shifted.

Six multiplied by one, add three to get this one, six multiplied by two, add three to get this one, six multiplied by N, any number, add three, is my position-to-term rule or my Nth term.

What are the position-to-term rules for these sequences? And then I want you to think about true or false: To find the Nth term rule for a sequence, all we need to know is the first term and the difference between the terms. So pause the video to have a go at that.

Absolutely excellent job if you managed to get the answers to this one.

Hopefully, you recognised that the difference, or the multiples that we're going up in, are fours, but that's being shifted one down So we got four n subtract one And that's the position-to-term rule.

For this one, I can see that we're going up in tens.

So my multiples of ten, but I've subtracted 13, and we're thinking about negative numbers as an extra burden if you managed to get that by subtracting 13, to get down to negative three.

Remember, if you're finding negatives a little bit tricky, just remember to try and use a number line.

So if I thought I've got ten, and I want to get all the way to negative 3, I need to subtract ten to get to zero.

And then subtract another three, to get to negative 3.

So altogether, I subtracted 13.

Now this was a tricky question.

To find the Nth term rule for a sequence, all we need to know is the first term and the difference between the terms. True of false? Well let's have a think about it.

If I didn't know any of these numbers here, but I was told the difference was four, I'd know I was dealing with multiples of four.

So I'd know that we've got 4n.

I would also know, if I got my first term as three, that the first term of my multiples of four, is four.

So I just need to subtract one, so yes I can get the Nth term.

That is true.

Just from the difference and the first term.

Excellent job if you managed to get that and if you managed a reason as well that's fantastic.

Pause the video now to have a go at your independent learning task.

So your independent task look like this on your worksheet, and you were using this grid to answer some of the first question, and these grids for the second question.

So let's see if you got the answers correct.

Which column in the grid to the right has the following position-to-term rules: Five n Five n, we're just looking at multiples of five So we can see that it's column D.

Five n subtract three, we've just moved those multiples of five three spaces to the left.

Down three spaces, so that got us to column A.

And the same thing for c.

For number two, we were using these grids here in these columns that have been highlighted for us.

To work out the position-to-term rules for columns a,b,c and d.

I know that in this column, I know straight away, what do I know straight away? Hopefully that I've got multiples of four because I've got four columns.

So I know it's four n and then to get from four n or the multiples of four, to this column I go down one, to this column I go up two, so that's where these have come from.

This one, I know I've got seven columns So in multiples of seven, seven n.

To get to negative four though, I have to go down 11.

So that's a bit of a tricky one.

Seven n to get to negative one, I had to go down eight.

For question three this should say.

The position-to-term rules for the following sequences.

So this time, we don't have the columns to help us, but you could've drawn a grid if it did help you.

I know my difference is two, so I've got multiples of two, and my multiples of two, two, four, six, eight, etc.

I've got to go down one to get to this sequence.

Similar for the other ones.

Well done, especially if you got this one.

We got some negatives in there which always makes it a bit trickier.

We're going up four each time.

So I've got multiples of four n that you get from my multiples of four, to the negative five negative one three, I've actually shifted it down nine spaces.

So again, well done if you used the number line for that, that's brilliant.

Now we're onto our Explore task.

So I would like you to look at the descriptions of these three arithmetic sequences.

You got three different people here saying three different things.

Some of the terms in my sequence are prime numbers.

The first term of the sequence lies between zero and negative five.

There are four terms between zero and 30.

I would like you to find Nth term rules that you can think of to match each of these descriptions first of all.

Now there are loads and loads for each description.

In fact, I think almost infinite numbers, maybe not, well yeah, with the prime numbers too.

So, make sure you are only writing one or two examples at the most.

I would start off by writing out the sequences, and then finding the Nth term of those sequences.

It's quite tricky to find the Nth term straightaway.

Once you've done that one, so we'll label that question one once we've done question one, I would like you to have a go at question two.

Can you find any Nth term rules that would work for all three descriptions? So for all three of those apply to one sequence, can we do that? And my little hint for this one would be to start with this person, when you're trying to find the sequence that works for all of them.

Pause the video now to do that.

So as I said, there were loads of examples for the first question.

Here are just some that worked for me, that I found.

So I know that two is a prime number, so I can just start with two, and then write any old sequence I wanted to.

I decided to go up in threes.

And then I worked out the Nth term of three and subtract one.

For this person Xavier, I decided to start with a number between zero and negative five, any old number, and I picked negative two.

And then I just decided to go up in twos.

And I can see that my final answer is two n subtract four And for Binh, I have decided to, well in fact this was quite an interesting one, because for this one it was a little bit more, had to think about it a little bit more.

So I wanted four terms between zero and 30.

So my first thought was I've got a difference of 30 divided by four, because I've got four terms, I know I'm roughly going to need about seven between seven and eight as a difference between each term.

So I picked eight n and then I just found one that worked.

This was the start of my thinking to find the answer for number two.

There are actually two possible answers: Seven n subtract nine, or eight n subtract 11.

So if you managed to get seven n or eight n you can then shift them to make them work for this person and this person.

Cause we just needed a prime number, which is quite easy to get in your sequence somewhere, and the first of them to be between zero and negative five.

So massive well done, especially if you answered number two cause that was quite a tricky one, and it might have taken you a bit of time.

So really well done for that.

That brings us to the end of today's lesson.

So do share your work again with your class teacher, and share your work with Oak National, if you'd like to.

Please ask your parent or carer to show your work on Instagram, Facebook, or Twitter tagging @OakNational and #LearnwithOak.

Really well done with today's lesson again.

You've been fabulous, and I'll see you for the next lesson on sequences.