Lesson video

Hi, I'm Mr Chan.

And in this lesson, we're going to learn about simple quadratic and cubic sequences.

Let's begin by looking at this sequence.

Do you know what sequence of numbers is shown? That's right, you might recognise this sequence as the squared numbers.

So one squared is one, two squared is four, three squared is nine, four squared is 16, etc.

Well done if you spotted that.

So in this example, we're asked to generate the first five terms of the sequence n squared plus three n plus two.

I know that that's quadratic because the highest power of n there is a two, that makes the n squared a quadratic sequence.

So let's begin by looking at the position.

The first five terms that means we're looking for the term in position one, the term in position two, the term in position three, etc.

We're going to start off by breaking the sequence n squared plus three n plus two into its component parts, into the n squared first, then the three n, then the two, and we'll add it all together at the end.

So let's look at the first number in the n squared sequence, that would be one.

The second number in the n squared sequences four.

So again, these are the squared numbers one, four, nine, 16, 25, as you can see.

Now three n means that we've multiplied the position number by three.

So we'll get one times three, two times three, three times three, etc.

So we get three, six, nine, 12, 15.

The add two parts of the sequence just means we're just adding two constantly.

So we add two, add two, add two.

Now in terms of putting all that together, to generate the first five terms of the sequence, n squared plus three n plus two, we would add those values together there.

So we're adding the n squared part, the three n part, and the two part altogether.

One, add three, add two, we would get six.

Continue with the second position, the second term in this sequence, we would add the four, the six and the two, to get 12.

Similarly, with the others, nine add nine, add two, we would get 20.

16 add 12 add two, we would get 30.

And finally the fifth term in the sequence we would get 42.

So the first five terms of the sequence would be six, 12, 20, 30, 42.

Here's a question for you to try.

Pause the video to complete the task, resume the video once you're finished.

Here are the answers.

So when you use diagrams to show squared numbers, they indeed do look like a square, as you can see in this first question.

So you generate the squared numbers by doing one squared, to get One, two squared, four, three squared equals nine, four squared would be the next term on the fifth term in the squared numbers would be 25 that's five squared.

Here's another question, please try.

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Here are the answers.

This question illustrates really interesting facts about the squared numbers.

And what happens from term to term in the squared numbers.

You'll notice that the diagrams help really well with this type of question.

And what you can see happening is that the squared numbers increased by consecutive odd numbers.

So we get add three, then add five, then add seven, then add nine, etc.

Let's see if you can generate the next few.

Here's a question for you to try.

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Here are the answers.

So hopefully in this question, you used the table to help you generate the sequence n squared plus two n takeaway one.

So you find the if you fill in the terms individually into the table, and then add the group of terms that you need to generate the sequence, you will in fact get the sequence two, seven, 14, 23 and 34.

Here's a sequence of numbers.

Do you know what sequence this is? Yes, these are the cube numbers.

Cube numbers are generated by repeatedly multiplying the number by itself.

So we start off with one.

So we get one cubed, which means one, multiply by one, multiply by one to get one.

Two cubed, that would mean two times two, times two, to the eight.

Three cubed 27, four cubed 64, five cubed 125, six cubed 216.

And we carry on like that.

Let's generate the first five terms of the sequence n cubed plus n squared plus two.

I know this is a cubic sequence because it's got an n cubed in the nth rule.

So let's have a look at how we do that.

Again, we're looking at position one, two, three, four, five.

These will be the first five terms. And we're going to break up the sequence n cubed plus n squared plus two into smaller powers.

So we're going to look at the n cubed first, then the n squared, then the two and add all that together at the end.

So let's look at n cubed.

Now n cubed simply means we're multiplying the position number by itself three times.

So we get the cubic sequence like this.

One cubed, two cubed, three cubed, four cubed and five cubed.

So that's one, eight, 27, 64, 125.

Let's get the n squared sequence next.

So that's the squared numbers.

One squared, two squared, three squared, four squared, five squared, one, four, nine, 16, 25.

And the two just means we're adding two constantly at the end, so we just add two like this.

Now in order to put all that sequence together, so the sequence n cubed plus n squared plus two, we would add those three parts together, one, add one, add two, to get us four.

The second term in the sequence would be eight, add four, add two, to get 14.

Third term, adding those parts together 27 add nine, add two, to get 38.

And the last two terms in a similar fashion 64 add 16, add two to get 82.

And the fifth term 125 plus 25 plus two, 152.

So the first five terms of this sequence n cubed plus n squared plus two would be four, 14, 38, 82, 152.

Here's some questions for you to try.

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Here are the answers.

So in question four, it hasn't given you a table to help you this time.

But there's nothing stopping you drawing your own.

So you'd need to work out n squared first in part a and then add on five n.

In part b, you'd have to work out n squared, and then double n squared to get two n squared and take away three to generate the sequence.

So hopefully you got those correct.

Here's a question for you to try.

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Here are the answers for question six.

Again, this question hasn't helped you with giving you a table for generating these cubic sequences.

So there's nothing stopping you drawing your own to help you.

But the first question is just the cube numbers add five because that's n cubed add five.

And in part b, it's the cube numbers, subtract the squared numbers.

So generating those putting them in a table and then subtracting the squared numbers from the cube numbers should give you the sequence zero, four, 18, 48, 100.

I hope you got those correct.

Here's another question please try.

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Here are the answers.

So in this card matching question, the only one that may have proved a little bit tricky is the card three bracket n squared plus two.

Now there's a couple of options you could do with this sequence here.

You could have expanded bracket to get three n squared plus six.

Or you could have worked out the sequence n squared plus two and then multiplied it by three.

Whichever way you did it, you would have got sequence nine, 18, 33, and 54.

So hopefully you got that correct.

That's all for this lesson.

Thanks for watching.