Lesson video

Well done for choosing to learn using this video.

My name is Miss Davies and I'm gonna help you as you work through the lesson.

With that in mind, it might be really useful to be able to pause bits, really think about some of the concepts that we are talking about, and I'm gonna help you by adding in any suggestions or any hints that might help you with all the different tasks we're gonna be working on.

Let's have a look at this lesson then.

Welcome to today's lesson on generating a sequence, using a position-to-term rule.

By the end of the lesson you'll be able to appreciate that a sequence can be generated and described by a position-to-term rule.

A couple of key words that you may be familiar with, each value or pattern in the sequence is called a term.

A term-to-term rule describes how to calculate the next term from the previous term.

A position-to-term rule, which is what we are focusing on today, describes how to generate the term from the term number, which is a different type of rule to a term-to-term rule.

We're gonna have a look at writing position-to-term rules and then generating terms from position-to-term rules.

As we've mentioned, each value or pattern in a sequence is called a term.

Each term has a term number so we can refer to specific terms. The term number is the position of the term in the sequence.

So if I have the sequence 3, 5, 7, 9, the first term is three.

So we say that the term number is one 'cause it's the first term.

If we're using our shorthand notation, we can write that as T with a subscript of one, T1 equals three.

There's lots of other notations that can be used, this is the one we are gonna use today for efficiency.

The second term is five, so the term number is two.

We can also write that as T2.

The third term is seven, so we call the term number three and we can write that as T3 equals seven if we wish.

Let's look at this pattern.

How could you describe the pattern for the number of lines in each term for this sequence? You could have said something like, each term adds three more lines.

That would be a term-to-term rule 'cause you're describing how to get from one term to the next, adding three extra lines.

Let's look at how we could use the position-to-term rule.

We need to look at the relationship between the term number and the term.

So let's put our term numbers above our terms, and then to make it easier, I'm gonna write down the number of lines.

So it's 3, 6, 9, 12.

Notice that the term number is the number of triangles in the pattern.

So term 10 will have 10 triangles in its pattern, and that's how we can refer to the term numbers.

What is the relationship then between the number of triangles and the number of lines? What do you reckon? You might have said something like each triangle needs three lines, or the number of lines is the number of triangles multiplied by three.

To make that into a position-to-term rule, we need to look at the relationship between the term number, and the value of the term.

So how do we get from the term number one to the number of lines three, or the term number two, to the number of lines six? So we could rewrite our statement about triangles to say that the number of lines is the term number multiplied by three, then that's a position-to-term rule 'cause we're linking the term number, to the term.

If the position-to-term rule is the number of lines, is the term number multiplied by three, how many lines will there be in the 10th term? What do you reckon? Well done if you said 30 lines, 'cause we can do the 10th term, so 10 multiplied by three.

How many lines will there be in the hundredth pattern? Well done if you said a hundred multiplied by three or 300.

Using a position-to-term rule is a quick way to find any term from its term number.

If you've done any work with term-to-term rules, you know it could be quite time consuming to find larger term numbers.

With a position-to-term rule, we've got this nice quick way to get a term from its term number.

How could we calculate a position-to-term rule for the number of lines in each term for this sequence? We'll go through it together.

So again, let's put our term numbers across the top, and just like previously, the term number is the same as the number of triangles.

So for the 10th term I'll end up with 10 triangles.

I'd like you to pause the video and have a think about what the relationship is between the number of triangles and the number of lines.

Off you go.

This relationship was a little bit trickier to spot, so well done if you said something like each triangle needs two extra lines, except the first triangle needed three lines.

So you could have said that it's two lines per triangle, plus one for that first one.

If we write this as a position-to-term rule, the number of lines is the term number multiplied by two plus one, and that's a rule to get from the term number, to the term.

We can now check that rule works for each term.

So the term number times two add one, gives us the number of lines.

The term number times two add one, gives us the number of lines, the term number times two add one gives us seven which is the number of lines.

If we do that for the last one, four times two is eight plus one is nine.

So that is our term.

Let's see if we're happy with this idea then.

So match the sequences on the left to the position-to-term rules on the right, and these position-to-term rules are talking about the number of lines in each pattern.

It may help for you to count the lines and write them down.

See if you can get them matched up.

Well done.

Let's have a look.

So the top one is the number of lines is the term number multiplied by three plus one.

Second one is that bottom one, the number of lines is the term number multiplied by four plus one.

The one that looks like some sort of H letter H's, the number of lines is the term number multiplied by three plus two.

So that's second one down.

And then the bottom picture matches with the third description.

The number of lines is the term number, multiplied by four.

We've looked at some patterns.

Let's have a look at numerical sequences.

How can we describe what is happening in this sequence? 4, 8, 12, 16.

You might have said something like the sequence is adding four each time, or you might have said the sequence is the four times table.

If you said the sequence is adding four each time, that's a term-to-term rule.

It's telling you how to get the next term from the previous term.

So let's see if we can find a position-to-term rule.

So what then is the relationship between the term number and the term? So the first term is four, second term is eight, third term is 12.

Well, the term is the term number multiplied by four.

This is our four times table.

This is another sequence that's adding four each time.

but it's not the four times table.

So 6, 10, 14, 18.

Again, we'll write our term numbers.

We know our term-to-term rule would be add four, and that means that we can write each term by adding fours onto the same value, every term must be something add four, add four, add four.

Let's look at this first term then, how could we write that as something add four? Well six is two add four.

Our second term then 10, is six add another four.

So two add four, add four.

14 is gonna be six add four add four.

So we've got that as two add four, add four, add four.

Let's have a look at this structure in some more detail.

How can we write T10 in the same format? Well done if you spotted that that's gonna be two, add four, add four, add four, and you should have 10 additions of four there.

Let's see if we can spot a simpler way to write this.

Remember multiplication is the same as repeated addition.

So we've got two add four, add four, add four 10 times.

We can write that as two plus 10 lots of four, or two plus 10 times four.

Looking at this pattern can help you generalise what is happening.

So T1 we said was two plus four, and that's like saying two plus one lot of four.

T2 was two plus four plus four, or two plus two lots of four.

T3 ended up being two plus three lots of four, and T4 two plus four lots of four.

Finally T10 is two plus 10 lots of four.

Hopefully you have spotted that the term number is the number of fours we're adding up.

So the value of a given term is two plus the term number multiplied by four, or multiply the term number by four and add two.

Let's look at another sequence.

What is the same as what is different from our previous sequence? This sequence also has a term-to-term rule of add four, but it starts on a different value.

Again, we can write each term by adding fours onto a set value.

So we could write 20 as 16 plus four, couldn't we? Then T2 would be 16 plus four plus four.

T4 would be 16 plus four plus four plus four plus four.

What do we notice then about the number of times we add four to get each term? Well for the first term it's 16 plus one lot of four, the second term 16 plus two lots of four and so on.

And it follows the same pattern that we had before.

So the number of times we add four is the same as the term number.

The position-to-term rule would be multiply the term number by four and add 16.

So all Aisha says is the position-to-term rule for the sequence 6, 11, 16, 21, 26, could be multiply the term by five, then add six.

How could we check if Aisha is correct? Well we can try it out.

We can use the term number, use her rule, and see if it gets us to the term.

One time five plus six is 11.

Two times five plus six is 16.

That is not the sequence she has written.

So her rule does not generate the correct term from its term number.

Where do you think Aisha has gone wrong? We'll work it out together.

So she needs to start by writing T1 as a value add five.

She's definitely spotted that there's something to do with five, but she needs to write the number six as something add five.

So let's write it as one add five.

T2 then is one add five add five, and T3 is one add five, add five, add five.

Now we can see that you multiply the term number by five, but you only add one.

Let's use this method to work out the position-to-term rules.

There are other methods as you develop your sequence of skills for working out position-to-term rules that you might find easier to do.

Okay, we're looking at this example 'cause we're picking apart the structure of these sequences.

So let's start by working out what we're increasing by each time.

We've got a constant additive rule of plus three.

Let's write T1 as something plus three then, so five is two plus three.

T2 then is two plus three plus three.

And T3 is two plus three plus three plus three.

We've got a pattern now.

We can see that we've got two plus one lot of three for T1, two plus two lots of three for T2, and two plus three lots of three for T3.

We can say that that the term is the term number multiplied by three, add two.

Can you use the same idea to work out the position-to-term rule for the sequence 13, 23, 33, 43? Off you go and then we'll look through it together.

Well done.

So we know we're adding 10 to get the next term.

So let's write T1 as a value add 10.

So 13 is three add 10.

So that means 23 is three add 10 add 10.

And now we can write those using our multiplication.

So it's three plus one lot of 10, then three plus two lots of 10, and so on.

So our rule could be the term is the term number multiplied by 10, add three.

And we could check that works for the fourth term for example.

So 43, is that four times 10 add three? Yes it is.

Well done.

Time to have a practise then.

So for each sequence you need to write a position-to-term rule for the number of dots in each pattern.

It may help you to count the number of dots and record it in the table before you try and write your rule.

Give it a go.

Question two, four students are writing a position-to-term rule for the sequence 9, 11, 13.

Who is correct? For the incorrect answers, can you show me how you know they are incorrect? Have you got a way of checking? Off you go.

Well done, we're really starting to bring those sequence ideas together now.

So A, you might have said something like the term number multiplied by five, add one.

For B, you might have had the term number multiplied by four, add three.

For C, the term number multiplied by six, add one.

And D, the term number multiplied by four, four lots of the term number.

So Alex was correct, the term is the term number multiplied by two add seven, and that is correct.

Nine is one lot of two add seven, 11 is two lots of two add seven, and 13 is three lots of two add seven.

Let's see if we can show that the other three are incorrect.

So Sam, we can try the first one.

So the term is the term number multiplied by nine, that works for their first number, one times nine is nine.

But it doesn't work for their second number.

Two times nine is not 11.

Let's look at Sophia's then.

So she said the term is the term number add two, I think she's got in a muddle.

The term-to-term rule could be add two, but it's not the term number add two that gives you the term, because one plus two is not nine, two plus two is not 11, three plus two is not 13.

So the term number add two is not the correct rule to get from the term to that term number.

And then Izzy's, the term is the term number multiplied by two, add nine.

Well let's try it.

One times two is two, add nine, that's not nine.

The second term, two times two is four, add nine, that's not 11.

So again, her rule does not work.

Let's have a look now then at generating terms from a position-to-term rule.

If we know the position-to-term rule, we can generate any term in the sequence.

That's why position-to-term rules are so useful.

Aisha says the position-to-term rule for my sequence is multiply the term number by five, then subtract two.

Izzy's questions quite interesting.

How do I know what number to start on? Well if Izzy wants to work out the first term in Aisha's sequence, then the term number is one.

So if she wants to know what to start on, if she wants the first term, you could do one multiplied by five, then subtract two and that would give you the first term.

So Aisha's sequence starts on the number three.

How could Izzy work out the second number in this sequence now she knows the first one is three? Well we're gonna use the same rule, we multiply the term number, so for the second term, that's two, multiply by five, subtract two.

So that gives us eight.

So Aisha's sequence starts three, eight.

See if you can work out the next three terms in her sequence in the same way.

At the moment we might not know how the sequence grows.

However we can use the term number to generate T3, T4, T5, and then we can sort of see how the sequence grows.

So if we do three times five take away two, it's 13, and four times five take away two is 18.

Five times five take away two would be 23.

So our sequence is 3, 8, 13, 18, 23.

"If I wanna work out the 10th term, this might take me a while", Izzy says.

Is there a quick way for Izzy to work out the 10th term? What do you reckon? Brilliant, the whole point of a position-to-term rule is you can get straight from the term number to the term.

So all we need to do to find the 10th term is 10 times five take away two, which is 48.

The position-to-term rule can help us investigate types of sequences that we're not as familiar with.

Izzy's come up with a really interesting sequence.

What about if I cubed the term number and add one? What will that sequence look like? I'm not sure at the moment.

Let's see if we can work it out.

So one cubed plus one is two.

Two cubed is eight plus one is nine.

Three cubed is 27 plus one is 28.

Four cubed is 64 plus one is 65.

Five cubed is 125 plus one is 126.

So our sequence is 2, 9, 28, 65, 126.

That's a really interesting sequence and it's not obvious at the moment how that sequence is growing.

However, we can use the idea of a position-to-term rule to find any term in that sequence even if we can't see a pattern between the terms. What would the 10th term be? We can just do 10 cubed plus one.

So 10 cubed is 1,000, plus one is 1,001.

It's time for you to have a check.

Can you match the position-to-term rule with the 10th term? Off you go.

Well done if you've got all of these correct.

So the top one matches with the bottom, 10th term, so T10 is 46.

The second one matches with the third, so T10 is 68.

The third one matches with the second one, so T10 is 103.

And the bottom one matches with the top one, T10 is 110.

Working out position-to-term rule first can help us find larger term numbers in some types of sequences.

So what would the 50th term be in the sequence that starts 12, 21, 30, 39? Well we can write the term numbers and then the terms underneath, and then we can find a position-to-term rule to get from the term number to the term.

We can see that this has got a constant additive pattern of plus nine.

That would be a term-to-term rule.

If I want the 50th term, I'm gonna have to keep adding nine on lots of times, which is gonna be time consuming.

So maybe we can find a quicker way.

Well let's work out a position-to-term rule.

So that first term of 12, we could write as three plus nine.

That makes the second term 21, three plus nine plus nine.

And that third term three plus nine plus nine plus nine.

The position to term rule would be multiply the term number by nine and add three.

For the 50th term then we could do 50 multiplied by nine, add three, and that gives us 453.

Andeep says, the sequence 4, 7, 10, 13 and sequence 7, 10, 13, 16, have the same position-to-term rule.

Jun then says that means they have the same 10th term.

I'd like you to pause the video and see if you agree with either of those statements.

I wonder if you notice then that Andeep is incorrect.

Even though both sequences go up in threes, and they have lots of the same terms, they start on different terms. That means they are different sequences and different sequences have different position-to-term rules.

Jun is correct that if they have the same position-to-term rule, they would have the same 10th term, but these sequences can't have the same term.

The second sequence will have a larger 10th term, won't it? And that just goes to show that Andeep's original statement that the sequences are the same, have to be incorrect.

They are different sequences, they have different position-to-term rules, and therefore a different 10th term.

So now we're gonna work through one together and then you're gonna have a go at one.

So watch what I do for this first one.

What is T100 for the sequence which starts 30, 37, 44, 51? Now the easiest thing to do will be to work out a position-to-term rule first, and then I can work out my hundredth term.

So we can see that this is going up by seven each time.

It has a constant additive pattern of plus seven.

We can write T1 as 23, add seven, and then T2 as 23, add seven, add seven.

The position-to-term rule will become multiply the term number by seven and add 23.

We can then use this to find the hundredth term.

So we can do a hundred, which is the term number multiplied by seven and add 23.

We end up with T100 being 723.

Okay, time for you to have a go at one yourself, and then we'll work through it together.

So what is T100 for the sequence which starts 12, 20, 28, 36? Pause and give it a go.

So let's start by working out a position-to-term rule.

We've got a constant additive pattern of plus eight.

So let's try and write our terms as something plus eight.

T1 we could write as four plus eight.

Then T2 would be four plus eight plus eight.

T3 would be four plus eight plus eight plus eight.

So our position to term rule becomes multiply the term number by eight and add four.

To find T100 then we could do a hundred multiplied by eight, add four, and that gives us 804.

As I previously mentioned, you might find some quicker and easier ways to write position-to-term rules as your sequences skills develop.

But we're really pulling apart the structure of these sequences to show how much we understand about these additive relationships.

Right a chance for you to have a practise then.

So for each sequence I'd like you to find the value of T60.

So the 60th term.

Some of them I've given you the position-to-term rule, others I've written the first few terms in the sequence and I'd like you to find the position-to-term rule yourself and then you can work out the 60th term.

Give those a go and then we'll have a look at our answers.

Well done.

Let's have a look at your answers.

So for A each term is the term number multiplied by five plus six.

So for T60, when you do 60 multiplied by five plus six, and we get 306.

B, the term number half, then add two.

So we're gonna do half of 60, which is 30, and add two, which is 32.

C, the term number subtract five, so that's gonna be 55.

And then D, 10 less than the term number, then multiplied by three.

So 60 subtract 10 is 50 multiplied by three is 150.

Right, these last three was a little bit of a challenge 'cause we need to find the position-to-term rule first.

So we can see that we're adding three each time.

So we can write each term as a value add three.

The first term is going to be nine add three.

Then we're gonna have nine add two lots of three, nine add three lots of three.

So it's the term number multiplied by three, add nine.

So 60 times three add nine, gives us 189.

Well done if you got that one.

Right F we can see that we're increasing by one each time.

So let's see if we can write each term as something add one.

Well 18 would be 17, add one.

Then 19 would be 17 add one, add one, and so on.

So this time we've got the term number multiplied by one, add 17, or the term number add 17.

So 60 add 17 is 77.

Really good job guys.

Let's look at this last one with decimals, but the method is exactly the same.

So we can see we're increasing by 0.

4.

So let's see if we can write every term as something add 0.

4.

So T1 would be one add 0.

4.

T2 would be one add 0.

4, add 0.

4.

And that pattern will continue.

So we've got one add term number multiplied by 0.

4.

So if we do 60 multiplied by 0.

4, we get 24, and then we can add one to get 25.

Be really, really proud of yourself for giving those a good go.

Even if you didn't quite get to the final answer, there's lots and lots of skills there.

Maybe just pause the video, see if you can work out where you've gone wrong, and then you'll be ready to carry on developing these skills in the future.

Well done.

Let's finish by looking at what we've talked about today.

So we can use the term number to describe a sequence using a position-to-term rule.

We can then use a position-to-term rule to generate any number in a sequence.

We talked about how it's a nice and easy to find any number in a sequence from its term number if we have that position-to-term rule.

Really good work today guys.

I look forward to seeing you again in the future.