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 where we're introducing quadratic sequences.

Our focus today is gonna be on recognising the features of a quadratic sequence.

This might be a type of sequence you've seen a bit of before or it might be completely new to you.

We're gonna talk a little bit about the nth term.

Now the nth term of a sequence is the position of a term in the sequence where n stands for the term number.

And you've probably seen nth terms for arithmetic sequences before.

We're looking at quadratic sequences.

We need to be comfortable with what a quadratic is.

Now quadratic is an equation, graph or sequence whereby the highest exponent of the variable is two.

So when we've seen this before with quadratic expressions, when x is the variable, they have general form a x squared plus bx plus c.

So we're gonna start then with identifying and generating quadratic sequences.

And in the second part of the lesson we're gonna be looking at some nth term rules for simple quadratic sequences.

Andeep is investigating a sequence which starts 3, 6, 11, 18, 27.

He says, "I want to predict what the next number will be, but I can't spot any patterns." Pause video, can you see any potential term-to-term rules for this sequence? Well, we can see that it's not arithmetic or geometric, but if you look at the differences, you can see that we're adding three than five than seven than nine.

Laura says, "The difference in the differences is two." Is there a better way for her to express this idea? Right so those values we looked at before.

The plus three, plus five, plus seven plus nine, that seem to form a linear sequence.

We can call that the first difference.

It's the difference between successive terms. The difference between the differences we call the second difference.

So this sequence has a common second difference.

Laura's changed her explanation now.

"This sequence so far has a common second difference of two." Andeep says, "We have seen this before with the square numbers.

There must be lots of sequences with a common second difference of two." There are, so it would be useful if we knew the nth term rule so we can generate this exact sequence.

We've seen that before with linear sequences.

If we know the nth term rule, we can generate any term in the sequence and we can solve some more complicated problems. Right.

Well, let's look at the square numbers.

So here's our first four square numbers.

What would be the nth term rule for the sequence of all square numbers? Yeah, of course we have n squared.

If you square the term number, you get the term.

And we can see, like we said before, that this does have a common second difference of two.

What would happen if we added two to each number? Let's have a look together.

So we add two to each number, therefore we'd have to add two to the nth term rule, and we get our new sequence, Which is n squared plus two.

What do you notice about this sequence? Right, just like the square numbers, it has a common second difference of two.

In fact, it's the same sequence we were looking at before, which Andeep started with.

So any sequence with a non-zero common second difference is called a quadratic sequence.

The nth term rule of a quadratic sequence will be a quadratic expression.

So we'll have general form an squared plus bn plus c, where A is not equal to zero, we do need a coefficient of n squared, otherwise it would be linear.

So we look at some examples, n squared, 3n squared, n squared minus three, 2n squared plus 4n minus one.

Anything with that general quadratic fall.

nth terms, which wouldn't give us a quadratic sequence, 2n, two to the power of n, n cubed, or zero n squared minus 2n plus nine.

If you want to read through what those sequences look like, pause a video and have a look at that now.

Laura says, "I think the sequence which starts 2, 4, 8, could be a quadratic sequence." What do you think? With only three terms, it's hard to say what a terms term rule might be.

You might have said it looks like it's gonna be geometric doubling each time, but we don't know with only three terms. In fact, without knowing the rule, we don't know if a sequence is gonna continue even if we had lots of terms. If this was a quadratic sequence, let's look at what the next term would be.

Will be between two and four, we're adding two, and between four and eight we're adding four.

So those differences are increasing by two.

So next time we'd have to add six, so we'd get 14, and those could be the first four terms of quadratic sequence.

Now the common second difference can be any constant value.

We've seen two a lot today, but we look at this quadratic sequence, it adds 2, then 6, then 10 and has a common second difference of four.

Right, quick check.

What is the common second difference for each of these if they were quadratic sequences? Off you go.

So adding one, then two, then three.

So we have a common second difference of one.

With B, we're adding 1, then 6, then 11.

So we have a common second difference of five.

Have a C, we're adding five, then seven, then nines.

So you've got a common second difference of two.

Right, which of these could be the first four terms in a quadratic sequence? C is the only one that could be.

You look at A, it's a linear sequence.

The common second difference is zero.

For B, you'll see that we're adding four, then five, then nine, which is a second difference of one, then four.

So not a common second difference.

The last one we're adding five and sixth and seven.

So it does have a common second difference of one.

Sometimes it can be quite difficult to identify if a sequence could be quadratic.

Some of these sequences have some really interesting forms. Writing out the first and the second differences is gonna be really helpful here.

Pause video.

Do you think these could be the first five terms in a quadratic sequence? And then we'll check together.

Well, we're subtracting two, then subtracting one, then adding zero, then adding one.

Well, to get from negative two to negative one, we add one.

Negative one to zero, we add one.

And from zero to one we add one.

So yes, that could be a quadratic sequence with a common second difference of one.

Even though the terms start by decreasing and then increasing.

It's really important when we write our first differences that we including whether we're subtracting or adding to get from one term to the next.

If we were to graph these values, you can see that they could be forming a parabola.

We know quadratic equations form parabolas, and the same would be true for quadratic sequences.

Now we can see how it's possible for a quadratic sequence to decrease and then start increasing again.

Andeep says, "Could a quadratic sequence have a negative second difference?" Yes, by definition, a quadratic sequence has a non-zero second difference.

This could include negative numbers or fractions.

Let's look at this example, 5, 9, 11, 11 9.

You can see there's a bit of symmetry.

We've seen equations of parabolas before that have followed this sort of pattern.

Let's check our differences.

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

Well, those differences are decreasing by two each time.

Here go, what would be the common second difference for this quadratic sequence? Let's have a look.

We're adding eight, then five, then two, the negative one.

It has a common second difference of negative three.

Pause video.

Which of these could be the first five terms in a quadratic sequence? Let's look them B and C could be forming quadratic sequences.

If we check A, we're adding one, then two, then three, then five.

Not a common second difference.

It could be a Fibonacci sequence.

For B, we're adding three, then two, then one, then zero.

So those first differences are decreasing by one each time.

Common second difference of negative one.

And for C, we're subtracting four, subtracting two, adding zero and adding two.

So common second difference of two.

Now if we know the nth term rule, we can generate the terms in a quadratic sequence.

Just like arithmetic sequences, we can substitute n equals one to get the first.

n equals two to get the second term and so on.

Let's look at the first four terms in the sequence n squared plus 10.

We have n is one, we'd have one squared plus 10, which is 11 and two squared plus 10, which is 14.

Three 3 plus 10 and 4 squared plus 10.

And we can keep doing that for any term.

We could check that we've still got a common second difference, and yeah we've got a common second difference of two this time.

These should be 10 more than the square numbers, and you can see that they are.

By your turn, what would be the first four terms in the sequence 5n squared, do you think? Right.

Well, five lots of one squared is five.

Five lots of two squared is 20.

Then we've got 45 and 80.

And again, if we check a common second difference, oh, we've got a common second difference of 10 this time, and this should be five times the square numbers, which we can see that it is.

Ooh, this one's a bit more interesting.

What would be the first four terms in the sequence n squared plus n? Let's have a look.

So we need substitute one everywhere that we see an n.

So it would be one squared plus one, and using brackets can help us get this correct, two squared plus two and so on.

We look at the common second difference.

This time we're adding four and six and eight, and it's a common second difference of two.

Right, Laura wants to find the first four terms in the sequence 3n squared minus two.

She finds the first term, which is one, and the second term, which is 10, she notices these have a difference of nine.

Once I have the first two terms, I can keep adding two to the difference and get the other terms. So now add 11 and add 13.

Ooh, what mistake has Laura made? This is similar to a method we sometimes use for arithmetic sequences.

Sometimes we get the first couple of terms and then if we know what the common difference is, we can form the rest of the linear sequence.

So Laura's trying to do something similar.

Unfortunately, not all quadratic sequences have a common second difference of two.

We've seen lots that do, but this one is gonna have a common second difference of six.

So just finding out that between 1 and 10, we're adding 9.

Not enough information to then carry this on.

Alright, see if you can help her out.

Can you find the correct third and fourth terms? Now you're gonna explore more and more quadratic nth terms and you'll start seeing patterns and getting more of an idea of what these sequences look like.

For now, let's just substitute n is three and n is four, should get 25 and 46.

And like I said before, we do have a common second difference of six this time.

Laura says, "I can use my calculated check as well." Of course we can.

Right, your turn.

For each question, I'd like you to work out the common second difference and then what is the next term in each sequence? Off you go.

Question three, which of these could be the first five terms in a quadratic sequence? And question four, I'd like you to use the nth term rules to generate the first four terms in each of these sequences.

Off you go.

And finally, some slightly more interesting nth term rules here.

Still all gonna be quadratic sequences.

F and G are in factorised form.

Give these a go.

So for A, we have a common second difference of one.

B, a common second difference of four.

C, a common second difference of two and D, a common second difference of one.

We'll look at the next term in a moment.

For E, a common second difference of negative four.

F, a common second difference of two.

G, a common second difference of negative one and H, also a common second difference of negative one.

We look at our next terms then.

Pause the video and check you've got those correct.

So the ones that could be forming a quadratic sequence, B, E, F, G and I.

Well done.

Right, I'd like you to pause the video and check your first four terms for each of these sequences.

You might have found that you could use one of your previous sequences to help you get the next sequence.

For example, if you already have n squared to get n squared plus six, you just need to add six to each term.

If you already have a half n squared, if you want half n squared plus 10, you just need to add 10 to every term.

Check those and then we'll look at the last lot.

Then question five again, pause the video and check you've got all the correct terms. G was quite interesting, it was the difference of two squares, so it could be written as n squared minus nine, so nine less than the square numbers.

Make sure that you are happy with those.

And then we'll look at the next part of the lesson.

So now we're gonna start looking at the nth term rule for simple quadratic sequences.

And we've done most of the hard work for this already.

So we've seen how the square numbers have a common second difference of two, and we know that adding a constant to each term does not affect the differences.

Let's just show that again with n squared plus one.

If I add one to each term, I've translated the sequence by one and the differences remain the same.

Not only does it still have a common second difference of two, it's still adding three, then five and seven, then nine.

We can use this idea to help us find the nth term rule.

The sequences of the form n squared plus c, where c is a constant.

This is a really important point.

So you might wanna jot this down.

If a quadratic sequence has a common second difference of two, then the coefficient of n squared in the nth term rule will be one.

And that's why we've seen so many sequences with a common second difference of two.

So I'm gonna show you an example.

So we're gonna find the nth term rule for the quadratic sequence, which starts 10, 13, 18, 25, 34.

The first thing I'm gonna want to do is check for a common second difference.

The common second difference is two.

So this must be related to the sequence n squared.

Just like with arithmetic sequences, we can write out n squared and then compare it to our sequence.

So there's the sequence n squared.

Let's write our sequence below it and see how we get to our sequence from n squared.

We'll look at this one together.

You add nine to the square numbers to get our sequence.

So the nth term rule will be n squared plus nine.

We could check if a specific term follows this rule.

So let's try five squared plus nine.

Well that gives me 34.

So the fifth term should be 34, which it is.

Sofia wants to find the nth term rule of this quadratic sequence.

She starts by finding the common second difference.

What does this tell her about the nth term? Can you remember? Right well, we know it's gonna be a quadratic nth term and the coefficient of n squared will be one.

Sofia says, "If I add five, I get the square numbers.

So the nth term rule is n squared plus five." Do you agree? Alright, she's made a bit of a mistake here.

This sequence is five less than the square numbers.

So the nth term rule is n squared minus five.

What's gonna help is if she wrote the sequence of n squared first and then her sequence underneath, and then she'll see that she's subtracting five from each term.

Right I'm gonna show you how to do one and then you are gonna have a go yourself.

So finding the nth term rule for this quadratic sequence.

Let's look for a second difference.

I know the coefficient of n squared then will be one.

So let's look at 1n squared.

Well, there's the sequence n squared, and I'll write my sequence underneath.

And if I compare each term, I can see that I'm adding 34.

So my rule will be n squared plus 34.

I'd like you to use the same method to work out the nth term rule for this quadratic sequence.

Give it a go.

Right, you might be wondering why we bother finding the common second difference first, if we know it's gonna be related to n squared.

For some trickier nth term rules, it won't be related to n squared.

It might be 2n squared or 3n squared.

So it's good habits get into.

Here we can see that the common second differences two, so it is related to n squared and then we're adding 18 to each term.

So you should have n squared plus 18.

Let's try this one.

So again, check my common second difference.

There's n squared and my sequence underneath.

Writing it that way it means I'm gonna get this the right way around because to get from n squared to my sequence, I need to subtract six.

So n squared, subtract six.

Try it for this one on the right.

So we have a common second difference of two.

We write out n squared.

Write your sequence underneath and you can see we're subtracting one.

We've got n squared minus one.

Right, once we have found the nth term rule, we can use it to solve problems. So here's a quadratic sequence.

What is the 10th term in this sequence? Andeep says, "We can find the common second difference and keep generating terms until we've got the 10th." What would be a quicker way? Finding the nth term and then we just need to substitute n is 10.

So this one does also have a common second difference of two.

So if I look at n squared and then my sequence, I'll see that my sequence is seven more than the square numbers.

So if I want the 10th term, I just need to do 10 squared plus 7, which is 107.

Sofia says, "I have a quadratic sequence, which starts 2, 5, 10, 17, 26.

I want to know if 50 is in this sequence." And it does have a common second difference of two.

What could Sofia do? Right, this is exactly the same skill that you may have seen before with arithmetic sequences.

We find the nth term rule.

This sequence is n squared plus one, and then we just need to see if there's a value for n, which gives us a term of 50.

If we assume 50 is in the sequence, we can form the equation n squared plus one equals 50 and then solve.

So n squared would be 49, so n would be seven.

Laura says, "Don't forget the negative root n could be negative seven instead." All right, let's write that down.

So let's look at what this tells us.

It tells us that 50 is the seventh term in the sequence.

The term number cannot be negative.

So Laura's right, we don't need to worry about the negative root this time.

However, it is good practise to calculate all solutions and then explain why you are disregarding one.

Andeep is trying to justify if 46 is in the sequence with the nth term rule n squared plus 18.

There's his working.

What does he need to do to improve his answer? Right, we're justifying, we need to be really clear as to why 46 is not in the sequence.

So from n squared equals 28, we could find the square root of 28 and write it as plus or minus two, root seven or plus or minus 5.

29, and then n must be a positive integer.

So 46 is not in the sequence, and then we've explained why we've come to that conclusion.

Or you could have said 28 is not a square number because five squared is 25 and six squared is 36, therefore 46 is not in the sequence n squared plus eight.

It's just making sure that we are properly justifying our answers here.

Right so I've used, give this a go.

For each question, I'd like you to work out the nth term rule if these were forming a quadratic sequence.

Off you go.

Question two, we've got the first five terms of quadratic sequences.

I'd like you to work out the 10th term, see how you get on.

Right, and this time I'd like to work out the term number for the given term in each sequence.

So what is the term number? Off you go.

Then this question is all about the sequence with the nth term rule n squared minus 50.

I want to know if negative 50 is a term in the sequence you to justify your answer.

Do the same for 50 and 32.

And finally, what is the first term in this sequence greater than a hundred? And again, can you make sure you are justifying all your answers? Sofia wants to find the nth term for a quadratic sequence, which starts 3, 8, 15, 24.

She says the nth term rule is n squared plus two.

Can you explain how you know she must be incorrect.

For six, a quadratic sequence has consecutive terms, 25, 33, 44, 58.

They're not the first four terms, they're just four consecutive terms. Explain how we know the nth term cannot be of the form n squared plus c, where c is a constant.

Once you've thought about those ideas, we'll run through this together.

Right, check your answers for question one.

You should get n squared plus two, n squared minus three, n squared minus 11, n squared plus 2.

5.

Obviously if you've written that as five over two, that's absolutely fine as well.

For question two, finding the nth term first will be helpful.

And then for A, we've got 102.

B, is 127.

C, is 84 and D, is 99.

6.

Pause the video if you need to look over that again.

For three, if we set up an equation, n squared minus 10 equals 26.

n would be six or negative six, but we only want the positive answer here, so it'd be the sixth term.

For B, we do exactly the same thing, but 26 would be the fourth term in that sequence.

For C was forming an equation and solving 87 would be the ninth term.

And for D, we've got n squared to 625.

Well done if you calculated the square root as 25 or negative 25 but remember, we don't need the negative root this time to the 25th term.

For question four, we need to look at our justifications.

So n square minus 50 equals negative 50.

That gives us n square to zero and n is zero.

The n must be a positive integer.

Our first term in this sequence is when n is one.

For B, if we solve the equation, we get n is 10.

So it's the 10th term, so it is in this sequence.

For C, we get n squared equals 82, and then we need to explain why 32 then is not in our sequence.

I said because 82 is not a square number.

9 squared is 81 and 10 squared is 100.

So 32 is not in the sequence.

You might have done that a different way.

For D, different methods are acceptable.

You could solve an equation and then work out which value of n you need.

I've done a little bit of trial and improvement to help me get this right.

So 12 squared minus 50 is 94.

13 squared minus 50 is 119.

So the 13th term is the first one greater than 100.

And finally, Sofia must be incorrect.

The easiest way to show this is to just check the terms two squared plus two is six.

So the second term in n squared plus two would be six, not eight.

Question six, when a constant is added to the square numbers, the differences stay the same.

So the sequence should increase by three, then five and seven, then nine with a common second difference of two.

If we check this sequence, it has a common second difference of three.

So the coefficient of n squared in the nth term rule will not be one.

You may well explore some of these sequences where the coefficient of n squared does not one in a future lesson.

Well done for all your hard work today.

When we start a new topic or a new idea, there's always lots of exciting things to discover.

If you wanna to pause the video and just read through what we've learned today, our main focus has been this new type of sequence called a quadratic sequence and starting to explore some of the features.

You can also find the nth term of some simple quadratic sequences.

I hope you're excited to then extend that and look at some trickier quadratic sequences in the future.

I really look forward to seeing you again.