Lesson video

Hello, my name is Miss Parnham, and in this lesson, we're going to learn how to find the nth term, of a quadratic sequence.

We'll start the lesson by generating the first five terms, of the sequence with nth term n squared.

In other words, the square numbers.

So if we draw a table, to help us lay the workout, we have n which is the term number going from one to five, cause we want the first five terms. And the next row is n squared.

So we're going to multiply each one of those numbers by itself, and here we see the square numbers.

The first difference is the difference between, consecutive terms in the sequence.

So we can see the difference between one and four is three, between four and nine is five, between nine and 16 is seven, and between 16 and 25 we have a difference of nine.

Now that creates a sequence itself.

And the second difference is all about the difference between these numbers, we have just formed.

And as you may notice, this is two every time.

So the difference between three and five, the difference between five and seven, and the difference between seven and nine.

So let's investigate this a little bit further, We will try now n squared plus n.

Again, we'll do a grid, this helps us organise, and the first row will look exactly the same, because this is the term numbers every time, and the second row will also look the same, because this is based on n squared, and then we have the square numbers, but this time we're going to add n to those.

So one plus one, four plus two, nine plus three 16 plus four, and finally, 25 plus five.

And this is where we get the sequence two, six, 12, 20, and 30.

And just like before, we'll find the first difference.

And that's the gap between the consecutive terms, in that sequence.

So from two to six, from six to 12, from 12 to 20, and 20 to 30, we get the sequence of four, six, eight, and 10.

And just like before, we will look for a second difference, so that's the difference between four and six, which is two between six and eight is two, and between eight and 10 is two.

So what do we notice here? In both situations, two different sequences, both starting with n squared, we have the second difference of two.

Let's look at this a little further, so this time we're going to generate the sequence of three n squared.

We'll start exactly as before, with n in the top row, it's always that we have n squared in our second row, I'm going to multiply this by three.

So each of those square numbers is multiplied by three, and we look for a first difference.

So the difference between three and 12, between 12 and 27, between 27 and 48, and from 48 to 75, and here we have nine, 15, 21 and 27.

And when we investigate the second difference, that is always six.

Let's look at another example that involves three n squared, here we have three n squared plus 10n.

So we have a table just as before, we're going to go directly to three n squared because we can bring those numbers from the table above.

So we have three 12, 27, 48 and 75, and to those numbers that we are going to add 10n.

So to three we're going to add 10, to 12 we're going to add 20, to 27 we will add 30, to 48 we will add on 40, and finally 75 we will add on 50.

let's look for a first difference.

So that's the difference between 13 and 32, then the difference between 32 and 57, and then from 57 to 88, and finally from 88 to 125.

And this gives us this sequence of numbers looking for a second difference, we get six, six again and six again, so when we had sequences with nth term beginning with three n squared, the second difference was six.

Here's some questions for you to try pause the video to complete the task and restart the video when you're finished.

Here are the answers, you didn't have to draw a table to record your sequence and differences, but at the very least you was evenly spaced the numbers of the sequence and leaving enough gap between them.

So it's easy to write the first difference underneath each one, and then the second difference underneath that.

Here's some more questions for you to try, pouse the video to complete the task and restart the video when you're finished.

Here are the answers, making the link between the second difference on the n squared coefficient in the formula for the nth term is essential, for finding nth term of a quadratic sequence.

And that is why I wanted you to focus the first part of the lesson on practising this.

We're going to find the nth term of this sequence, and we can definitely see it's not linear, because it isn't increasing or decreasing by a constant amount.

We will know definitely that it is quadratic, when we get to find a second difference.

And if that is a constant amount, we know definitely it is a quadratic sequence.

So let's put this sequence into a table, tables aren't compulsory, but they do help organise our work.

So we'll look for the first difference and that's between zero and nine, so we have nine from nine to 22 is an increase of 13, from 22 to 39 we have a difference of 17, and from 39 to 60 It is 21.

So our second difference is the difference between these values.

So from nine to 13 is four 13 to 17, also four, and from 17 to 21 also four.

So we have a constant second difference, this is a quadratic sequence, and we know to have the second difference, to find the n squared coefficient.

So the n square coefficient is half of four, in other words two.

So this sequence is based on two n squared.

And two n squared is the square numbers doubled, so that's two eight, 18, 32 and 50.

What we need to do now is subtract that from our sequence, to work out the rest of the sequence, which will be linear.

So we're going to subtract two n squared, from our sequence.

So zero subtract two gives us negative two, nine subtract eight, which is one 22 subtract 18 is four 39 subtract32 is seven and 60 subtract 50 is 10.

So the rest of the sequence, once we have got the quadratic part is linear.

Let's now find the difference, that we normally would with a linear sequence.

And we can see that this is three each time.

So we're talking about a sequence based on the three times table, so we know it contains three n and three n looks like this three, six, nine, 12, and 15, compare that with the rest of sequence that we just generated, and you can see that we're subtracting five.

So putting all those clues together, we have an nth term two n squared plus three n subtract five.

Here's some questions for you to try pause the video, to complete the task and restart the video when you're finished.

Here with the answers, in part of did you notice that the pattern is a larger square, with two smallest squares removed from diagonally opposite corners? So the first square is a two by two square, with one square taken from the diagonally opposite corners.

So we can think of it as the pattern number plus one, all squared subtract two in algebra, that would be n plus one in a bracket, all squared subtract two.

If we expand that bracket, we get n squared plus two n plus one, and then we have the subtract two.

So simplifying that we gets is n squared plus two n subtract one.

Here's some more questions for you to try Pause the video to complete the task and restart the video when you're finished.

Here with answers, in question six, you can probably see from the diagram that the nth term is n squared plus four.

So every number in the sequence is far more than a square number, and we know 204 is far more than 200, but 200 Isn't a square number, so 204 is not in the sequence.

And if you substitute to find n and n does not turn out to be an integer, then the number cannot be in the sequence because n is the pattern number, and all the pattern numbers are integers.

That's all for this lesson.

Thank you for watching.