Lesson video

Thank you for joining me for today's lesson.

My name is Ms. Davies, and I'm gonna be guiding you as you explore some of these new and exciting sequences that we're looking at today.

Make sure that you've got everything you need before you start watching this video.

It's always a good idea to have a pen and paper so that you can jot things down and explore things in your own time.

Let's get started then.

Welcome to this lesson on quadratic sequences.

By the end of this lesson, you'll be able to find the nth term formula for any quadratic sequence.

If you need a reminder as to what the nth term formula is, or a reminder as to what a quadratic expression is, pause the video and run through those keywords now.

So we're gonna split this lesson into three parts.

We're gonna start by looking at how we can build sequences from other sequences.

Then that skill is gonna help us in the second part of the lesson, where we're looking at finding the nth term of sequences of the form N squared plus BN plus C.

And then we're gonna extend that to look at sequences of the form AN squared plus BN plus C.

So let's start then by building some sequences.

Here are the first five terms of two different linear sequences.

Pause the video.

What are their nth term rules? This skill is gonna be really useful to us today, so we're gonna take the time to check we know how to do this.

So you find the common difference, which is negative two.

So this sequence will be related to negative two N.

If I write out my negative two times table, then I need to see how I get to my sequence.

And you'll see that I need to add 12 to every number in the negative two times table.

So my sequence is, negative two N plus 12.

For the other one, we've got a common difference of four, so it's related to four N.

If I look at the four times table, I'm subtracting three to get my sequence.

So four N subtract three.

Well done if you got those yourself.

Now, we could add the first terms of the sequences together, then the second terms and so on to form a new sequence.

Do that now, what would the first five terms be? Right, we get 11, 13, 15, 17, 19.

That has nth term rule two N plus nine.

What do you notice about this nth term rule? Can you spot anything? Right, well you might have spotted that if we add the other two nth term rule, so negative two N plus 12, add four N minus three, we get two N plus nine.

If we add the nth term rules together, we get the nth term rule for the new sequence.

What this means is we can combine sequences by adding corresponding terms together, and the nth term rule of the new sequence will be the sum of the nth term rules of the other sequences.

And this is gonna be really, really useful.

So have a look at these linear sequences.

Which two of these could be added to get the sequence six, 13, 20, 27? Right, we could do this by finding the nth term of that sequence, which is seven N minus one.

So it could be three N plus four N minus one.

Now, what sequence would we get if we added two N plus five and five minus N? What do you think? So you could add the terms together, and see we get the sequence 11, 12, 13, 14.

If we add the nth term rules, we should get N plus 10.

And that is our sequence.

Right, now we've got the hang of this idea.

What do you think the new sequence would look like if we added a linear sequence to a quadratic sequence? Let's see.

So there's a quadratic sequence, N squared, which is our square numbers.

And underneath I've got the sequence two N plus three.

If I add them together, I get six, 11, 18, 27.

Now, let's look at the features.

So the new sequence is another quadratic sequence with a common second difference of two.

Alex says, "I think the nth term rule will be N squared plus two N plus three.

Sounds like a good guess.

How could we check? Well, we could substitute.

So if we try one squared plus two lots of one plus three, we get six, which is our first term.

Do the same for two.

We get 11, which is our second term.

We do the same for three.

We get 18, which is our third term.

So it's looking good.

So we've seen how adding the nth term rules of any two sequences will give the nth term rule of the sum of those sequences.

And this means we can build more complicated quadratic sequences by adding a quadratic and a linear sequence together.

So there's a quadratic sequence, N squared, and three choices of linear sequences.

Which two sequences can be added to make five, seven, 11, 17? See if you can figure this one out.

Well, we can check what sort of sequences this is first.

This is a quadratic sequence with common second difference two.

So it must be the sum of a quadratic and a linear.

So well done if you spotted it was gonna be N squared and five minus N.

What would the nth term rule of this sequence be? Right, we generally start with the highest exponent and then work towards the constant.

So, N squared minus N plus five.

Doesn't matter which way round you write those terms. Quick check then.

Which sequences could be added to make the sequence which starts eight, 14, 22, 32? And then tell me what the nth term rule would be.

Off you go.

This is a quadratic sequence, so it must be the sum of a quadratic and a linear.

So we're gonna start with N squared and then add one of the others.

And well done if you spotted it was three N plus four.

It's the only way we're gonna get the second term is 14.

It works for the third and the fourth term as well.

The nth term rule then would be N squared plus three N plus four, simple as that.

But I'd like you to use substitution then to check that this could be the nth term rule for our sequence.

Try it for each term.

We've got one squared plus three lots of one plus four, which is eight.

Do the same for two, and three, and four.

So we've looked at this a couple of ways now, and we've checked our answer, so you should be fairly confident that we've got this right.

Right, time to put this into practise.

I'd like you to write the first four terms of each of these sequences.

And then use those sequences, and tell me which ones could be added to make each of the sequences below.

Use that to write their nth term rules.

Have a play around with those.

Come back when you're ready for the answers.

Right, those should be your sequences.

So pause the video and check you've got those right.

If you didn't, that might have impacted your answers to question two.

So do check that now.

Right, so we've got three different quadratic sequences; N squared, two N squared, and negative N squared.

And then four linear sequences.

Let's have a look at which ones add to make A.

You should have sequence A plus sequence D, which gives N squared plus N plus two.

For B, it should be D plus E.

It is a linear sequence.

In this case, it's gonna be the sum of two linear sequences.

We get four N minus three.

For C, we've got A plus F, which is N squared minus N plus four.

Then for D, we've got B plus E.

We've got two N squared plus three N minus five.

You might have noticed that that has a common second difference of four, that sequence.

E, D plus G, that's another linear sequence.

Eight minus N or negative N plus eight.

And F, we've got C plus D.

Negative N squared plus N plus two.

You might have noticed that one's got a negative second difference.

And then the last column, we've got C plus E.

That was quite a strange looking quadratic.

The nth term, negative N squared plus three N minus five.

H was B plus F, which is two N squared minus N plus four.

And this one is actually three N squared.

Right, so now we're gonna apply everything we already know to finding the nth term of some sequences.

So identifying key features of a quadratic sequence is gonna help us start.

So let's have a look at this example.

You got six, nine, 14, 21, 30.

Now, these could be the first five terms in a quadratic sequence.

How do we know? Well, we're gonna look at the differences.

We're adding three, then five, then seven, then nine.

Has a common second difference of two.

Sam says, "The coefficient of N squared in the nth term rule will be one." Absolutely correct.

The fact that the common second difference is two means that the coefficient of N squared will be one.

So we're gonna start by looking at quadratic sequences where the common second difference is always two.

Because the second difference is two, this must be related to the sequence N squared.

So what we can do is write out N squared, write out our sequence underneath, and compare them.

What do you notice? So our sequence is five more than the square numbers.

Its nth term rule then is N squared plus five.

Right, let's look at a different sequence.

You might wanna jot this one down.

What are the second differences? So we're adding six, then eight, and 10, then 12.

Slightly different to the previous example, but we've still got a common second difference of two.

This means that the coefficient of N squared in our nth term rule will be one.

We're still comparing this to N squared.

Alex has noticed what I noticed.

"But this time it doesn't start by adding three then five." Let's see what this means.

So compare our sequence to N squared, what do you notice? Right, we have to add three to get our first term.

Ooh, then add six, then add nine, then add 12, then add 15.

There is not a constant translation between N squared and our sequence.

We can't write this as N squared plus a constant this time.

Our sequence can be made by adding N squared and the sequence three, six, nine, 12, 15.

So what is the nth term rule of the sequence formed by those differences? Right, it's the three times table, so three N.

So the nth term rule for our sequence is gonna be N squared plus three N.

You can now find the nth term rule for any quadratic sequence with a common second difference of two using that same method.

Let's look at a different one.

This time we're adding five, then seven, then nine, then 11.

This still has a common second difference of two.

This means the coefficient of N squared will be one.

So we can compare our sequence to N squared again.

This time we need to add one to get our first term, add three to get our second term, five, then seven, then nine.

What linear sequence then needs to be added to N squared to get our sequence? Right, those differences have an nth term of two N minus one.

So if we add N squared to the sequence two N minus one, we get our sequence.

So our nth term rule must be N squared plus two N minus one.

We're gonna put that into practise.

So I'm gonna show you the layout of one on the left, you're gonna try one on the right.

We still need to check our common second difference.

It is two.

Which means that the coefficient of N squared in our nth term rule will be one.

So let's look at one N squared and then look at our sequence.

I need to add one, then add five, then add nine, then add 13.

I need the nth term rule then for these differences.

So one, five, nine, 13 is increasing by four.

It's four N minus three.

The overall length term then it must be N squared plus four N minus three.

Of course, I could check a few terms, particularly if I have my calculator handy, to make sure that works.

Right, I'd like you to do the same then for the quadratic sequence, which starts eight, 13, 20, 29.

So we've got a common second difference of two, so this is related to N squared.

It doesn't start plus three plus five plus seven, so it's probably not N squared plus a constant, but we'll see that.

So coefficient of N squared will be one.

And compare our numbers, so we have to add seven, then nine, then 11, then 13.

It's the sequence two N plus five.

So our nth term is N squared plus two N plus five.

And again, you can always check a few terms with your calculator.

Alex is finding the nth term of this quadratic sequence.

This is a strange looking sequence.

We've got negative one, negative two, negative one again, two, seven.

He's found his second difference, and he's compared his terms. So the nth term rule is N squared plus four N minus two.

Can you spot the mistake he has made? He's written the square numbers underneath his sequence.

And sometimes it's easy to do this by mistake, particularly if you you're trying to save space on your paper.

Really, he wants to do it the other way around.

'Cause what this has meant is that he's calculated the differences incorrectly.

He needs to see what is added to the square numbers to get his sequence, not the other way around.

Right, he's gonna do it the other way now.

So to get from the first square number to the first term, we add negative two, then negative six, then negative 10, negative 14, then negative 18.

What is the nth term of the sequence added to N squared to get our sequence? Right, it's negative four N, 'cause the values are decreasing by four each time, plus two.

We've got N squared minus four N plus two.

The trickiest bit with this method is the negative number skills, and making sure you're subtracting things the right way and you're remembering that decreasing linear sequences have a negative coefficient of N.

We can check.

Check for our first term, we should have negative one.

Check our fifth term, we should have seven, and we do.

Right, let's try that again.

So I'm gonna give you an example on the left, and then you are gonna have a go.

So I have a common second difference of two.

So the coefficient of N squared will be one.

Let's look at N squared and then look at my sequence.

I add one, then I add zero, then I add negative one, then I add negative two.

Those values are decreasing by one, so it's gonna be negative N.

But that would start on negative one, and I wanted to start on one.

So negative N plus two.

My nth term is N squared minus N plus two.

Try your one.

Right, we have a common second difference of two.

We're adding zero, then two, then four.

The coefficient of N squared will be one, so let's write out N squared with our sequence underneath.

Right, and take our time here.

So to get from one to one, we're adding zero.

From four to one, negative three.

From nine to three is negative six.

From 16 to seven is negative nine.

They're decreasing by three, so it's gonna be negative three N.

I need it to start on zero, so negative three N plus three.

We've got N squared minus three N plus three.

Well done if you've got that one.

We're looking at the trickiest possible examples of these at the moment.

Right, time put it into practise.

I'd like you to find the nth term rule for each of these linear sequences.

And then what sequence needs to be added to N squared to get each of the sequences below? Use that to write their nth term rules.

So we're building sequences just like we did in part one of this lesson.

Give those a go.

Right, here are the first four terms of some quadratic sequences.

Can you find the nth term rule? And you've got four more to do.

And you've got four more quadratic sequences, thinking about your decimal and your fractions skills.

You could convert them all into fractions or convert them all into decimals if that helps you.

Give those a go.

Let's have a look at our answers.

So for A, you should have five N.

B, two N minus two.

C, three N plus two.

D five N minus eight.

E, seven minus N.

And F, 10 minus four N.

So in order to get our quadratic sequences, we need to do N squared plus D, which gives us N squared plus five N minus eight.

N squared plus B, which gives us N squared plus two N minus two.

N squared plus E, which gives us N squared minus N plus seven.

Then N squared plus A, which gives us N squared plus five N.

N squared plus F, which gives us N squared minus four N plus 10.

And finally, N squared plus C, which gives us N squared plus three N plus two.

And now bring that all together.

So for three A, we've got the sequence N squared plus N.

B, we've got N squared plus N plus three.

For C, we've got N squared plus three N plus four.

And for D, we've got N squared plus two N minus six.

Pause video if you want to look at working out in more detail.

For E.

Nice one this one, N squared minus 10.

For F, N squared minus two N plus eight.

For G, we've got N squared minus five N plus five.

It was an odd looking sequence, that one, 'cause we've already got the first four terms. For H, N squared minus three N minus two.

And then look at our fraction skills.

We can write this as N squared plus 0.

5N plus one, or N squared plus a half N plus one.

For B, we've got N squared plus N plus 0.

5, or N squared plus N plus a half.

The differences are increasing by one each time, they're just fractional.

For C, we've got N squared plus 1.

5N minus two, or N squared plus three over two N minus two.

And finally, N squared minus 0.

5N plus 0.

5, or N squared minus a half N plus a half.

We bring lots of our skills together there.

Our linear nth term squeals, our quadratic nth term skills, our negative number skills, and our fraction skills.

So if you have made some mistakes there, just spend some time going through and seeing what skill it was that you found a little tricky.

Right, now we're looking at sequences of the form AN squared plus BN plus C.

Now, we've seen sequences of the form N squared plus BN plus C have a common second difference of two.

There's an example.

What is the common second difference in the sequence two N squared.

Try this one out.

Right, we get a common second difference of four.

Make a prediction.

What do you think the common second difference will be for three N squared? Let's try it out.

We've got a common second difference of six.

So that's what we've seen so far.

What conjecture could you make from this information? You might have gone with something different.

I said the coefficient of N squared in the nth term rule is half the common second difference.

Now, we're not able to test every single sequence to see if this is true, so we're gonna try and show this algebraically.

Now, this is quite an interesting way of showing why this works.

So follow along as best as you can, and then we'll talk about the conclusion and how we can use it.

So any quadratic sequence can be written in the form AN squared plus BN plus C, but A, B, C are constants and A is not equal to zero.

When N is one, this is A plus B plus C.

When N is two, we get four A plus two B plus C.

When N is three, we get nine A plus three B plus C.

And when N is four and N is five.

Pause the video if you want to check you're happy with those.

What we can now do is write a generic sequence with these as our first five terms. So any quadratic sequence will have first five terms of the form A plus B plus C, four A plus two B plus C, nine A plus three B plus C, and so on.

Right, let's look at the differences.

So to get from the first term to the second term, I need to add three A and B.

Then I need to add five A and a B.

Then seven A and a B.

Then nine A and a B.

As expected, we can see a linear relationship in the differences.

Let's look at the second difference.

To get from three A plus B to five A plus B, we add two A.

And again, and again.

So we've got a common second difference of two A.

But remember A is the coefficient of N squared.

Hence we've shown for a sequence AN squared plus BN plus C, the coefficient of N squared is half the second difference.

And because we've shown that, we can now use it.

Jacob has multiplied the square numbers by constant A to get the sequence AN squared.

The common second difference is 10, so what is the coefficient of N squared? Well done if you said five.

If you half the common second difference, you get the coefficient of N squared.

Write the first five terms of the sequence five N squared, and verify that we do get the correct common second difference.

Right, there's N squared, and sometimes writing N squared first can help you.

So there's five N squared, and we do have a common second difference of 10.

We can now use this to help us find the nth term of any quadratic sequence.

Let's look at this one together.

So let's start by finding the common second difference.

We have a common second difference of eight.

So if we half that, we get the coefficient of N squared.

So the nth term is in the form four N squared plus something.

So what we need to do is write out the first few terms of the sequence four N squared.

Now, sometimes writing N squared first and then times in each by four can be helpful.

What can we do next, do you think? Right, I'm gonna want to write my sequence underneath and compare them.

So I'm subtracting three, and three, then three.

Oh, that's helpful.

Each term is a translation of negative three.

So the nth term rule is four N squared minus three.

Let's have a look at another example.

Start by finding the common second difference.

We need to half the common second difference to get the coefficient of N squared.

So this is related to two N squared.

Let's write out the first few terms of two N squared and compare it to our sequence.

Subtracting two, then adding one, then adding four, then adding seven, then adding 10.

So we've not added the same constant to each term this time, but there is a linear sequence.

So a linear sequence has been added to two N squared to get our sequence.

We just need the nth term of that sequence.

Negative two, one, four, seven, 10 is increasing by three each time.

Gives three N minus five.

The nth term then is two N squared plus three N minus five.

Right, I'm gonna do one more example for you and then you are gonna have a go.

So I need to find my first differences.

And my second difference, it's 12.

This means the coefficient of N squared will be six.

Let's write out six N squared, and then my sequence underneath.

And I'm subtracting nine, subtracting eight, subtracting seven, and subtracting six.

It's a linear sequence with nth term N minus 10.

My nth term rule then is six N squared plus N minus 10.

Right, you give this one a go.

So your common second difference should be six.

Your coefficient of N squared will be three.

There's our value to three N squared.

There's your sequence.

We need to add one, then three, then five, then seven.

Which is the sequence two N minus one.

Well done if you've got three N squared plus two N minus one.

Right, Sam has picked a really tricky sequence.

They want to find the nth term of this quadratic sequence.

But they've got this far, and the second difference is negative.

What do you think Sam needs to do now? Right, exactly the same as before.

We're gonna half the common second difference and get negative two.

So we should have negative two N squared.

And again, it might be helpful to write out N squared or write out negative N squared first to help you get negative two N squared.

And then look at our sequence.

What's the nth term of the linear sequence? Yeah, three N plus six should have negative two N squared plus three N plus six.

Right, let's have a practise.

I'll do one and then you can have a go.

We're subtracting five, then subtracting seven, then subtracting nine.

Those differences are decreasing by two each time.

The coefficient of N squared will be negative one.

So let's write negative N squared, and then our sequence.

And we need to add five, but then add three, then add one, then add negative one.

That's negative two N plus seven.

So we've got negative N squared minus two N plus seven.

Try it for your sequence.

We need to subtract two, then subtract eight, then subtract 14.

Those differences are decreasing by six.

The coefficient of N squared will be negative three.

So let's look at negative three N squared.

And your sequence, and we need to add 17, then 24, then 31, then 38.

There's lots of arithmetic in these questions, so take your time to get it right.

That's seven N plus 10.

So we've got negative three N squared plus seven N plus 10.

Laying out your work really clearly will help if you make a silly mistake with adding or subtracting.

Right, time to put that all into practise.

Part of four different sequences with nth term rule of the form AN squared is given below.

So it's just the square numbers multiplied by a constant.

By finding the common second difference, determine what A must be for each sequence.

Give it a go.

Your turn.

Now, here are the first four terms of some quadratic sequences, can you find the nth term rule? And you've got four more to do.

Give these ones a go.

And Jacob is trying to find the nth term of the quadratic sequence which starts three, five, eight, 12.

You can have a look at his working.

He says, "I don't think this one can have a quadratic nth term as the second difference is not divisible by two." Write a sentence explaining why Jacob is incorrect.

And can you find the nth term for this sequence? Give that challenge a go, and then we'll look through the answers.

Let's have a look.

We've got a common second difference of 10.

So this must be five N squared.

For B, a common second difference of six, so three N squared.

And C, common second difference of one, so 0.

5N squared or a half N squared.

Keep that in mind for the last question, that's gonna help us.

Right, hopefully you've got some beautiful working out to go with your answers.

Pause the video and check those four.

Now, check your answers for E, F, G, and H.

G and H had negative common second differences, so those ones were possibly a little bit trickier, So well done if you got both of those correct.

So Jacob is incorrect.

Any sequence with a common second difference could be quadratic.

It is possible to half one, it's just gonna give us a decimal or a fraction.

So we haven't seen many examples of this yet, but it's absolutely fine.

We're gonna do exactly the same way.

So let's have a look.

We half our common second difference to get half, so we need to do a half N squared.

That's 0.

5, two, 4.

5, eight.

Then look at our sequence.

We're adding 2.

5, three, 3.

5, four.

That's a nice linear sequence there.

It's the linear sequence, a half N, plus two.

So our sequence is a half N squared plus a half N plus two, or 0.

5N squared plus 0.

5N plus two.

Exactly the same method, just got to use our fractions and decimal skills.

Right, well done.

Just as I promised, you are now able to find the nth term of any quadratic sequence.

There was lots of new information in that lesson, but hopefully you saw how we built it up from a really simple concept of building sequences by adding the terms. Take a look through what we've looked at today.

You should be really proud of how hard you've worked and really proud of this new concept that you've wrapped your head around.

Thank you for joining me, and I really hope you choose to learn using one of our videos again.