Lesson video

Hi, everyone, it's Ms. Jones here.

And today, we are going to be learning about linear relationships.

So, we're thinking about what we've looked at already in terms of equations and sequences, and thinking about how there's a link to each other, and what relationships and what generalisations we can make.

Before we begin however, please make sure you have a pen and some paper, that you have a nice quiet space to work if you can find one and that you remove any distractions.

Pause the video here to make sure you've got all of that ready to learn.

Okay, let's make a start.

So, Xavier and Yasmin have written some number sequences.

So, you can see, Xavier's here, 11, 15, 19, 23.

And Yasmin's here, - 7, -1, 5, 11, et cetera.

Xavier is saying, "What sequence would we get if we added our sequences term by term?" So, if we added each term together, what would be the sequence that we would get after that? Yasmin was saying, "I think we can work out any term in the new sequence we form." So, I would like you to use the answer to Xavier's question, so work out the new sequence to explore what Yasmin is saying.

Pause the video to have a go at that.

So, we can add or subtract existing sequences to create new ones.

So, I've got Xavier's and Yasmin's here.

And if we add each of those terms together, which hopefully you did, you can see that this is the new sequence that we get.

This has its own nth term, which we can see here.

The difference is 10.

So, it's 10n and subtract six to get to this sequence here.

So, 10n subtract six.

And we can notice, hopefully you've noticed, that actually the nth term of our two original sequences added together gets us the nth term of our new sequence.

So, we can find these new sequences through addition and subtraction in two ways, either by using each term, and adding or subtracting by each term, or actually using the nth term rule.

And if we think about Yasmin's statement, she was saying, "Oh, I think we can find any term in this new sequence." Well, if you've got the nth term of that new sequence, then yes, you can find any term in it.

So, for example, what is the eighth term in the sequence formed by adding 4n and seven and 6n subtract 13.

Pause the video to have a go at that.

Hopefully, you found the answer 74.

Because remember, that n tells us the position, or which term we're looking at.

So, if we want the eighth term, we can do 10 multiplied by n which is eight in this case, subtract six which gets us 74.

So, really well done if you got that answer.

Pause the video now to complete your independent task.

The first question gives you two sequences.

The first five terms of two different sequences here.

The question it's asking is what is the 10th term of the sequence formed by adding these two sequences? So, just like in the connect, we can do one of two things here.

We can either add each term together to create our new sequence, or we could find the nth term of both of them, and add those together to find the nth term.

It's often a bit more useful to find the nth term if you're looking for something like the 10th or the 20th, or even an even larger term in the sequence because otherwise, we're going to have to continue that sequence on for 10 terms or 20 terms. For the second question, it's quite similar.

What is the 10th term of the sequence formed by adding these two sequences? So, let's do this one as an example.

So, this time, we've been given the nth term, and we can just add those together.

So, we're left with -2n add three.

And if we want the 10th term, we do 2 lot, -2 lot sorry, of 10.

Which is -20 add three which is -17.

So, well done if you got that.

B is very interesting.

It says decide if four will occur in the sequence formed by adding these sequences.

So, this time, we want to know if four will even show up in this sequence.

And again, there's two ways that we could do this.

Either we could write out the sequence -2n and three, and write out lots of terms, and see if four is going to come up in it.

Or remember what we said about the position being n.

I need to know if four is a position in it.

So, what I can do, and really, really extra well done if you did this, is I want to find out what n is.

Because if n is an integer, a whole number, then that means it has a position in the sequence.

So, we can solve this just like you would have done before solving linear equations.

And we end up with n equals -1/2 and that is not an integer.

So, we can say four is not in the sequence because it does not have a position in the sequence when we solve to find the position.

So, here are all of the answers.

Amazing job if you managed to get most of those right.

And incredible job if you managed to write a really good answer for 2b 'cause that was a tricky one, well done.

Now to the explore task.

Here, we have a sequence, and we have the first five terms of a sequence.

What I would like you to do is write three sequences that could be added together to form the sequence above.

But there are some conditions.

So, the first set of sequences I would like you to find are three ascending sequences that would create this sequence here if added together.

The next would be two ascending and one descending.

And finally, two descending and one ascending sequences.

So, think about what we need for the coefficient of n to know whether they're ascending or descending.

And think about whether we need to use the nth term to help us here.

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

So, there are lots of answers we could have got here.

The first thing I would have done was to find the nth term of this sequence.

So, I can see that we're going up in sevens.

So, we've got 7n.

And to get from the multiples of seven to this sequence here, I need to subtract 16.

So, what we needed here was three sequences, or three nth terms that would make this nth term.

So, once we've clocked that, it's actually not too tricky at all.

Ascending sequences, we need to make sure we've got positive coefficients of n.

But it doesn't really matter what we use as our second part of the sequence, whether we're subtracting or adding anything, as long as we have them all adding up to make -16.

So, I've got 2n, a 3n, a 2n makes 7n.

A -4 and -10 and -2 makes -16.

The next one, we needed two ascending and one descending.

So, we just need to make sure we've got two that are positive coefficients of n, and one that was a negative coefficient of n that sums to make seven.

And then finally, two descending and one ascending.

So, one with a positive coefficient of n, and two with a negative coefficient of n that sum to make 7n subtract 16.

Really well done if you managed to get any of those, or come up with your own as there were lots of different examples.

Amazing job today.

Make sure you complete the quiz for this lesson just to check your understanding.

And again, a massive well done from me.

This was a really interesting topic, and you did really, really well, so well.