Lesson video

It's really nice to see you in this lesson today.

My name is Ms. Davies and I'm gonna be helping you as you progress through the lesson.

With that in mind, please feel free to pause things so you can really think about the concepts we are discussing.

Let's get started.

This lesson is on generating a sequence from a term-to-term rule.

By the end of the lesson, you'll appreciate that sequence can be generated and described using a term-to-term approach.

So you might already be familiar with the idea of a term.

So each value or pattern in a sequence is called a term.

That means a term-to-term rule describes how to calculate the next term in a sequence from the previous term.

We're gonna start with this idea of using a term-to-term rule As we've just discussed, each value or pattern in a sequence is called a term.

We often refer to individual terms in a sequence.

We're gonna use this language of term quite a lot today.

So if I had the sequence 1, 3, 5, the first term is one, the second term is three, the third term is five.

I could then refer to a later term such as the 10th term is 19.

For efficiency, we can use notation.

There's different notation you can use.

In this lesson we are gonna use this notation.

So we're gonna call the first term T1, the second term T2, and then say the 10th term would be T10.

The capital T means we're referencing a specific term and then the number in subscript tells us the term number.

Sequences could be generated using the rule which tells us how to find the next term.

These are called term-to-term rules.

They give you a relationship between the terms. So let's say a term-to-term rule is add five.

What would the next three terms be if we start on 124? Well, we can add five, add five, and add five again.

You get a 129, 134, 139.

Let's have a look at some other rules.

So this time we'll do add -4 to the previous term to get the next term and we're gonna start on six.

Then multiply the previous term by five to get the next term and multiply the previous term by two and then add three to get the next term.

Pause the video and see if you can work out the next couple of terms and then we'll look at them together.

Okay, adding -4, then you should have had 2, -2, -6.

Multiply by five.

If you find any of them a little bit trickier, you can multiply by 10 and then half and that'll be the same as multiplying by five.

We should get 30, 150 and 750.

Multiply the previous term by two and then add three to get the next term.

So doubling six gives us 12, add 3 gives us 15.

Doubling 15 gives us 30.

add three is 33, and then 69.

We're gonna have a look at the same thing, but just applying this to our decimal skills.

So if we're adding -4, multiplying by 5, and then that last one of doubling and then adding 3, what would the next few terms be? Pause the video and see if you can have a go at this one.

Well done.

We should have negative -0.

8, -4.

8, -8.

8.

Again, remember that tip of multiplying by 10 and then halving.

So 16, 80, 400, and the last one, 9.

4, 21.

8, 46.

6.

Which of the sequences follow the term-to-term rule, "Double the previous term to get the next term?" What do you think? Well done if you spotted that both the first one and the second one follow that rule.

You may have looked at some of the other rules, so you could have a term-to-term rule of add three for that third rule.

That bottom left one you could have suggested that has a term-to-term rule of half or divide by two.

That bottom middle one seems to have a term-to-term rule of add 11 and that last one again looks like it's multiplying by half or dividing by two.

Quick check then.

Have a go at matching the term-to-term rule with the different sequences.

Off you go.

Well done.

You may have spotted that they all start 1, 4, but all the rules are different.

Multiply by four each time is the bottom one.

Multiply by three then add one is the top one.

Add three, add five, add seven, add nine gives you that second one down.

We're adding two more than we previously added.

And multiply by two then add two gives you 1, 4, 10, 22.

Andeep's classmates are trying to guess his sequence.

So he has said the term-to-term rule is multiplied by 0.

1.

Here are their suggestions.

Are any of them correct? What do you think? Lovely.

You might have said something along the lines of all these sequences have a term-to-term rule of multiply by 0.

1.

But there's no way of knowing if they are the sequence Andeep is talking about without more information.

So let's see if we can get some more information.

Andeep has now said the term-to-term rule is multiplied by 0.

1.

My sequence has the number four as a term.

Okay, let's see what his classmates say now.

Do all of the sequences match the information given? Are there any that don't work? What do you think? Really good spot if you notice they're all gonna be correct, even Lucas's, because although he hasn't written the number four yet, he carries on his sequence with the same rule of multiply by 0.

1, then it will have the number four.

Andeep's given it another go.

The term-to-term rule is multiply by 0.

1 and he said T2, so the second term is four.

Can we now work out the first five terms in his sequence? What do you think they're gonna be? Right.

Let's see if we can apply a method to help us.

So I'm gonna write out the numbers in the sequence with gaps for the ones that I don't know.

So I don't know T1 or the first term.

So I'm gonna leave a gap.

I do know the second term is four and then I don't know the third, the fourth and the fifth.

I know my rule is multiply by 0.

1, so I should be able to get the third, the fourth, and the fifth term by applying that rule.

Well done if you've got the same as me.

All right, how am I gonna get the previous term? Well, we know that we're multiplying by 0.

1.

So to get back to a previous term, we need to do the inverse, which is divide by 0.

1.

What might help us here is to think about our reciprocals.

So multiplying by 0.

1 is the same thing as dividing by 10.

So actually we could say that top row is all the same as divide by 10.

That means to work back to a previous term, we can multiply by 10.

If that's true, dividing by 0.

1 is the same as multiplying by 10.

Sequences is a really good chance for you to start looking and consolidating this idea of inverses, especially with decimals, with fractions, and with negative values.

You should have got got a first term of 40.

Well done if you've got all five.

Right, Andeep is thinking of a new rule.

Now the term-to-term rule is add three, the third term is seven.

What are the first five terms of his sequence? We'll do this one together.

So write them out like we said before and we're gonna put seven as the third term.

Adding three gives us 10 and 13.

Subtracting three during the inverse at our previous values are four and one.

Right, how could we then use this rule to find the 10th term in the sequence? So we can carry on with the term-to-term rule of adding three each time until we get the 10th term.

I wonder if you can spot any issues with this method.

It works okay when we're trying to work out the 10th term, but if we're adding a slightly more complicated value or if we wanted the 20th term or the 50th term or the 100th term, this rule isn't gonna work very well.

We're a lot more likely to make a mistake if we keep adding on 300 times, okay? We are likely to go wrong somewhere and not end up with the right answer and it's gonna be time-consuming.

So term-to-term rules are great for seeing how a sequence develops, but do be aware that they're not always the best way of describing a sequence.

They have their strengths and they have their weaknesses.

Okay, let's look at this sequence.

It starts at the number three and adds four each time.

Let's think about how we could work out the 10th term.

We could add four and keep adding four until we get the 10th term.

But let's think about whether there's a way to make this process quicker.

Let's think about how many times we need to add four to get from T1 to T10.

Did you figure it out? Hopefully you worked out that we need to add four nine times.

To get from term one to term 10, it's nine additions of four.

Okay, well, we know that repeated addition is the same as multiplication.

So we can do nine lots of four, which gives us 36.

So that overall addition must be 36.

Doing 3 add 36 is 39 is quicker than us adding 4 nine times.

Let's try it with another sequence.

So we're gonna start at 10 and add six each time.

Let's work out the 10th term.

So each addition is going to be plus six.

We need to do that nine times to get from the first term to the 10th term.

So 6 x 9 is 54, so that overall addition is 54.

So 10 + 54 is gonna be 64 for that 10th term.

Laura's now gonna try and use this rule.

My sequence starts at seven and adds five each time.

How can I work out T10? Aisha says to her, if you do 10 x 5 is 50, you know you need to add on 50.

What is wrong with Aisha's statement? Pause the video and have a think.

Well done if you said something like this.

There are only nine additions between T1 and T10, not 10.

So she shouldn't be doing 10 x 5.

She should be doing 9 x 5.

Hey, Jacob's had a go as well.

He says 9 x 5 is 45, so T10 is 45.

What do you think to his statement? Brilliant.

Well done if you spotted that he started off well, but he needs to add 45 onto the first term to get the 10th term.

Right, Lucas has given it a go.

T10 is 7 + 45 equals 52.

Do we now agree with Lucas? Let's work it out.

So if the first term is 7, adding 5 a total of 9 times, 9 x 5 is 45, the 10th term is gonna be 52.

Yes.

So Lucas has figured it out.

Right, Laura's thinking of a new sequence this time.

My sequence starts at 3 and doubles each time.

You might wanna write the first few terms out for this one.

So start at 3 and doubles each time.

To calculate T10, I can do 3 + 9 lots of 2.

What is wrong with Laura's statement? Pause the video and see if you can put it into words.

Well done if you spotted this 'cause it's important.

That method only works for sequences with a constant additive rule 'cause we are using the fact that repeated addition can be simplified as a multiplication.

Her rule doubles each time so there is a constant multiplicative pattern.

We cannot use this method to find larger terms. Be aware then that using term-to-term rules to find larger terms for sequences that aren't constant additive sequences, so more complex sequences, it actually can become quite tricky or even time-consuming.

Let's have a check then.

True or false? To find the 10th term of any sequence with a constant additive rule, you can multiply the first term by 9.

What do you reckon? Well done for spotting that is false.

Think about the justifications then.

Which of these are true? Well done for spotting that it's the rule multiplied by 9 that's gonna tell us the overall addition and then we need to add that onto our first term.

All sequences with the term-to-term rule of multiply by 5 and subtract 2 will be identical.

What do you think? Well done.

That one is false.

Think about what justification fits your answer.

Perfect.

It's this idea that sequences that have different first terms will be different sequences, even if they have the same rule.

Time to have a practise.

So below, you have a cross number which is made from sequences written from left to right or from top to bottom.

Fill in this cross number according to the rules.

So for example, where it says across and it says B, you need to find B and that rule is for B across.

So starting at seven and moving to the right.

C across, that rule is going to be starting at where the letter C is and moving from left to right.

You're gonna have to fill in some of them before you're able to fill in the others.

When you are done, fill in the missing rule for A down.

That means starting at the letter A, but moving from the top of the page to the bottom of the page.

What will that rule be? Give that a go and then we'll have a look together.

Very good.

This second set of questions then, for each sequence you need to use the term-to-term rule to find the desired term.

For some of them, you might want to write out some of your terms to get an idea of how a sequence is developing, okay? You might be able to use some of the methods we've talked about today to make some of them easier.

Give it a go and then we'll talk through how we did it in a moment.

Well done.

Let's have a look at this cross number then.

I'm gonna read out the answers in the order I would've calculated it in.

So let's start with B across.

So 7, 1, -5, <v ->11, -17, -23.

</v> I'm then gonna fill in A down.

So I now know I've got -4 and 1.

That's an addition of plus five.

Then -4, 1, 6, 11, 16, 21, 26.

Now I've got those filled in.

I can have a go at some of the others.

So I'm gonna have a look at E down.

So you should have -11, 6, 23, 40.

And F down, -23, 6, 35, 64, 93, and 122.

Now that we've got some of those terms in, we'll be able to see if we've made any mistakes.

So let's fill in C across and that has a pattern of add 12.

So it starts with 4.

Then we should have 16, 28, 40, 52, 64, 76.

And finally, D across, 26, 50, 74, 98, and 122.

Hopefully you got them all right and that you spotted any mistakes you made as you worked through 'cause we need all these sequences to have a constant additive rule.

So the first one start on 2 and add 7 each time.

T3 is gonna be 16.

Start on 1 and multiply by 3 each time.

This might have been easiest if you wrote them out.

So 1, 3, 9, and T4 is 27.

Start on 6, double then add 10.

Again, probably easy if you write them down.

So double 6 is 12, add 10 is 22, double 22 is 44, add 10 is 54, double 54 is 108, add 10 is 118.

Well done if you go that one.

Starts on 5 and add -4 each time.

Want the value of T6, that's gonna be -15.

Starts on 10, multiply by 2 each time.

Again, it might be easy to write them down as you go.

You should get 640.

Starts on -4 and add 5 each time.

Find the value of T10.

So it's 9 additions of 5 and that gets you to 41.

The last one.

Start on 24 and add -2 each time.

So that's 24 additions of -2.

So that's -48 we're adding on and that gets you down to -24.

Well done if you got most of those correct.

We're now gonna look at finding term-to-term rules.

How do you think we could write a term-to-term rule for this sequence? Perfect.

Add 10 to the previous term to get the next term would work nicely.

How about this term-to-term rule? What do you reckon? Well done if you spotted that's adding -6 or subtracting 6 from the previous term to get the next term.

If we know a sequence has a constant additive pattern, we can find a term-to-term rule, even if we have missing terms. We're gonna have a look at some of those now.

So these sequences have a constant additive pattern.

How could we find the term-to-term rule? Well, we don't have successive terms, so we're gonna have to do a little bit more work.

What we need to know is what we add onto the first term to get the next term.

What we can do is start by finding the difference between terms that we do know.

So for example, T3 subtract T1 is a difference of 16.

Let's consider how many times our value has been added to T1 to get to T3.

Well, it's two additions, isn't it? From one to 17.

If overall it's an addition of 16, then each individual edition must be 16 divided by 2 or 8.

So our term-to-term rule could be plus 8.

We can then fill in some missing terms to check that it works.

So that would give us 9 then that works for 17, gives us 25 and it works again for 33.

How about this one where we've got three missing terms in between? Well again, we can find the overall difference, 59 - 23 is 36.

Then we need to think about how many additions we have.

To get from T1 to T5, it's four additions.

36 divided by 4 will tell us what each edition is.

Brilliant.

So our term-to-term rule could be plus 9.

Again, filling in some terms and checking it works is a nice way to check our answers.

Right, I'm gonna have a go at one and then you are gonna have a go at one.

So find a term-to-term rule for a constant additive sequence with T2 is 7 and T5 is 19.

I'm gonna start by writing them out with the missing terms and then I'm gonna find the difference between the terms that I do know.

So 19 takeaway 7 is 12.

That's an overall addition of 12.

Then I've got three additions.

So 12 divided by 3 is 4.

The term-to-term rule then is plus 4.

And I could do seven, add four, add four, add four, and check I get to 19.

Time for you to have a go.

Can you find a term-to-term rule for the constant additive sequence with T1 is 2, and T6 is 42? Give it a go.

Lovely.

Let's look at that one together.

So we'll write them out.

T1 is two and T6 is 42.

We find our overall difference is 40 and that's five smaller additions between the terms. 40 divided by 5 is 8.

So term-to-term is plus 8.

Again, we can do 2 + 8 + 8 + 8 + 8 =+ 8 and check we got to 42.

Let's look at some non-additive relationships.

So what could the term-to-term rule be for this sequence? Well done if you spotted it's multiply the previous term by 4 to get the next term.

How could we write the term-to-term rule for this sequence? Give it a go.

Good spot if you notice that we are multiplying by -1/2 or dividing by -2.

So Alex says the first number in my sequence is one.

Is this enough information to work out the term-to-term rule for Alex's sequence? What do you reckon? No, of course not.

There's any rule that could work with starting with the number one.

So now he has another piece of information.

The second number is two.

Is this enough now? Can we work out the term-to-term rule for Alex's sequence? Well done if you said something along the lines of we could work out a rule that would work but it might not be the rule that Alex wants to use.

What rules can you think of that work? Give this a go.

I wonder if you came up with any of the same ones as me.

So I went with add 1, multiply by 2, multiply by 5 then subtract 3.

There's absolutely loads you could have had.

So now we've got three numbers.

We've got one, two, and four.

Alex is pretty certain now then there's only one sequence it can be.

Do you think he's right? Right, he's actually not correct.

There are now a lot fewer rules that would work, but it is possible there's more than one rule.

And remember, sequences don't necessarily have to follow a set mathematical rule.

That's why having this idea of term-to-term rule so we know how our pattern is developing is really important.

You could have had multiply by 2, you could have had add 1, add 2, add 3.

Often when we get three terms, we will be able to describe a term-to-term rule, but that doesn't mean it's the only rule that would work for those values.

So have a go at this one.

Which term-to-term rule could describe the sequence that starts 2, 6? Which of those you think it could be? Well done if you spotted at the top one and the bottom one.

How about 3, 5? Have a read of those rules.

Which ones do you think are correct? Lovely.

Three of those are correct.

Add 2, we can have multiply by 2 and subtract 1, or we can have subtract two then multiply by 5.

Let's put all these skills into practise.

So we have another puzzle.

You need to fill in each row and each column so they form a sequence with a constant additive pattern.

So each column and each row is going to have a constant additive pattern.

Different rows will have different additive patterns.

When you think you filled them all in, can you write the term-to-term rule for row A and row B? Give that one a go.

Question two, describe a term-to-term rule for each sequence below.

So can you put into words how to get from one term to the next term? And then question three, Alex and Aisha are writing term-to-term rules for sequence where T1 is 5 and T2 is 15.

Alex says, "Me and Aisha have different rules, so one of us must be wrong." Can you put into words why Alex is incorrect? And then think about what rules they could have written.

Lots of thinking involved in that.

So give it a go and then we'll come back together and see what we got.

Lovely.

I'm hoping you enjoy some elements of this puzzle.

So check your answers against the ones written on the screen and then your two rules, A was the term-to-term rule of add 6, and then B, it was a term-to-term rule of add -3 or minus three or subtract 3, something like that.

Well done.

Pause the video and do check that you've got all the right values.

Four question two, you should have a rule along the lines of add 3.

For B, you could have had multiply by 2.

For C, you could have had multiply by 10.

D, you could have had add -8.

E, multiply by -3.

There are potentially other options that would've worked.

I just went with the ones I thought were the most obvious.

For F, you could have had a couple of them.

I find this one is a little bit more challenging.

So well done.

You could have had multiply by 2 then add 1, or you could have had add 3, add 6, add 9, where we're adding three more on each time.

The last question then, why is Alex incorrect? Your answer might have included something like, it is possible that multiple rules can apply to the same two terms. Some of the rules that you might have come up with, add 10, multiply by 3, multiply by 2 then add 5, add 10, add 20, add 30.

There's loads of ones that you could have come up with.

So well done.

Fantastic work today, guys.

So we have looked at how a term-to-term rule tells us how to generate the next term based on the previous term.

We can generate sequences from a term-to-term rule.

We saw that using a term-to-term rule to find higher term numbers can be time consuming.

We looked at some methods that might help, but we know that, that's not gonna apply to all types of sequences.

And finally, it's possible to describe a sequence using a term-to-term rule, but these rules might not be unique.

There may be other rules that work for the same values.

Brilliant.

Fantastic lesson today, guys, and look forward to seeing you again soon.