Lesson video

Hello, Mr. Robson here.

Welcome to Maths, how good of you to join me.

Today.

We're doing arithmetic sequences and their graphs.

If we can combine all of those things we know about arithmetic sequences with so much of what we know about graphs, we're going to become very powerful mathematicians indeed, I'm excited about this lesson.

I can see that you are too.

Our learning outcome is I'll be able to recognise that arithmetic sequence can be shown graphically and how this relates to the equation of a linear graph.

You'll also use this to find the Nth term formula.

There's a lot of exciting learning ahead.

I think you're gonna enjoy it.

Lots of key words we're going to hear, an arithmetic or linear sequence is a sequence where the difference between successive terms is a constant.

For example, 20, 30, 40, 50 is an arithmetic sequence, whereas 20, 40, 60, 160 is not.

The Nth term of a sequence is the position of a term in a sequence where N stands for the term number.

For example, if N equals 10, this means the 10th term and the sequence two parts to our learning.

Today we're going to begin by graphing a sequence.

Visual representations are a powerful tool in mathematics to see what an arithmetic sequence such as 2N is doing.

We can turn the term numbers and values into coordinate pairs and plot them on a grid.

Let's take those N values, substitute them into 2N and we get those term values.

We can plot coordinate pairs in the form term number, term value, IE, the N value first, the T value second.

So from this table of values for the sequence 2N, we'll be plotting the coordinate pairs, one, two, two, four, three, six and so on.

We end up with those five coordinate pairs.

Notice that the points align.

Arithmetic sequences are also known as linear sequences.

They increase or decrease in a straight line.

Drawing a straight line through all of our coordinate pairs confirms that we've plotted an arithmetic sequence correctly.

Andeep spots it linear pattern and says the two point fifth term is five.

What's wrong with this statement? Pause, tell a person next to you Or tell yourself, see you in a moment.

Welcome back and well spotted, five is indeed not a term in this sequence to N.

The sequence goes two, four, six, eight.

We bypass five, five's not in it.

In fact, there is no two point fifth term.

There's a first term, a second term, a third term, a fourth term, a fifth term.

Term numbers have to be positive integers.

When plotting arithmetic, also called linear, sequences, they form a straight line, but we don't draw a line through the coordinate pairs.

We might draw a line to confirm our points are correctly aligned, but it's only the T values of the coordinate pairs that are terms in the sequence.

Quick check you've got that.

True or false, you draw a line once you've plotted a sequence.

Is that true, is it false? Once you've decided, select one of the statements at the bottom of the screen to justify your answer.

Pause and do this now.

Welcome back, and well said.

It is indeed false and you justified that answer with we only plot the points when graphing a sequence so that visual there isn't quite right.

That arithmetic sequence should be graphed like so.

Another check for you.

How do we know that this is not the graph of an arithmetic sequence? Pause, see if you can spot why that one's not arithmetic.

Welcome back.

Well spotted, the points do not form a straight line.

Therefore it's not a linear sequence, not an arithmetic sequence.

The graph of an arithmetic sequence allows us to see the features of the sequence.

In this case, 2N, we can see that it's an increasing sequence.

We can also see there's a constant additive difference of positive two.

You know that when you look at your term values, you can also see it in the graph.

A constant additive difference of positive two between each term.

It's going to be important, that constant additive difference, so I need to check you can spot it.

What's the constant additive difference of this arithmetic sequence? Pause and work it out.

Welcome back and well spotted, it's positive three.

You can see it between each term on the graph.

We can use different methods to plot an arithmetic sequence.

One option is to populate a table of values and plot each coordinate pair.

For example, if we were plotting the sequence for N minus one, we could do some substituting.

When N equals one, the term is three.

When N equals two, the term is seven.

When N equals three, the term is 11.

We can keep going with that and populate the table like so.

We've then got five coordinate pairs to plot on this grid.

The coordinate pairs one, three, two, seven, three, 11 and so on.

That's a perfectly valid option.

A straight line confirms our points align, but we're just graphing the terms of this sequence, so we didn't leave that line there.

Quick check you've got that.

I'd like you to populate this table of values and therefore plot the arithmetic sequence 3N minus two.

Pause, take a moment to do that.

See you in a minute to check your work.

Welcome back, hopefully your term values came out like so.

One, four, seven, 10, 13.

You then plotted the coordinate pairs one, one, two, four, three, seven, et cetera, and you then checked that your points aligned so you know you are right.

Of course, we don't leave that line there.

We're just using it to check the points align.

Another option doesn't require a table of values.

Let's plot two N plus three and do it a different way.

By substitution, we can find the first term, when N equals one, T equals five.

Our first term is five, we can plot that point.

What else do we know? The sequence has a constant additive difference of positive two between successive terms. Remember, in an Nth term rule, the coefficient of N defines our additive difference, so if we know it's got a constant additive difference of positive two between terms, we can use that to plot the next term and the next term and the next term and in fact, every term from there on in.

A straight line tells us that points align, we know we've done that right.

Quick check you can do that.

Plot this sequence without using a table of values, four N minus four, pause.

Give it a go.

Welcome back.

A good starting point is the first term, in this case when, N equals one, T equals zero.

We can plot that point.

We then got a constant additive difference of positive four between successive terms, so you should have plotted points like so.

Straight line confirms that you've done this bit of maths correctly, but we don't leave the line there.

That is the graph, the sequence, four N minus four.

Whilst we don't draw a line when plotting an arithmetic sequence, it is a useful device for helping us to spot errors.

Here's some work that Izzy's done.

Can you see which term Izzy has got wrong? Pause, tell the person next to you or say it aloud to yourself.

Welcome back, well spotted.

It's that one that doesn't align, so 21 is not in the sequence.

It must be an error.

Now errors are not something to be afraid of in mathematics.

They're just one step closer to the right answer, so we will try again.

When N equals four, seven lots of four minus five.

Ah, that's 23, not 21.

Quick check you've got that.

Which term does not belong in this arithmetic sequence? Pause, pick out the wrong term.

Welcome back.

A straight line through the sequence shows us that 13 does not align with the others, so it must not be in the sequence, it must be the error.

Something different happens when we plot this arithmetic sequence, 14 minus 3N.

The first term is 11 and we can plot that coordinate pair.

The sequence has a constant additive difference of negative three.

The coefficient of N is negative three so we can plot the next terms in the sequence.

Have you noticed something? This sequence has negative terms. If our sequence has negative terms, we're going to need the fourth quadrant of the coordinate grid in order to plot them.

Our points align.

We know we've done this sequence correctly.

Quick check you can plot a sequence like that one.

I'd like you to plot the first five terms of this sequence, 11 minus four N, pause and go for it.

Welcome back, let's check your work.

The first term is seven.

The sequence has a constant additive difference of negative four, so our successive terms should be plotted like so.

The points align, we know we're right.

Practise time now, question one, you're gonna plot the sequence to N plus one.

You're gonna do it twice and you're gonna use two different methods.

In part A, you're gonna use a table of values and substitution, for part B you're gonna find the first term and then use the common difference of the sequence.

Pause and do this now.

Question two, I'd like you to plot these arithmetic sequences.

For A, you're gonna plot 3N minus one, for B, you're gonna plot 10 minus 3N.

You may use whichever method you wish.

For question three, I'd like to highlight two errors in this plotting of the arithmetic sequence, seven minus 2N.

For each of the two errors you spot, I'd like a sentence explaining why each is an error.

Pause and do this now.

Feedback time, let's see how we did.

Question one, part A, your tailor values should look like so, coordinate pairs should look like, so.

For part B, the first term was three.

Constant additive difference of two means the sequence grows as follows.

Your checking mechanism is to make sure your points align, but please don't leave that line on your answer.

Question two, two arithmetic sequences to plot, for 3N minus one, if you used a table of values, it would've looked like that.

Whichever method you use, your coordinate pairs should definitely have looked like that and you would check of course that they align.

That's your graph of the sequence 3N minus one.

Part B, 10 minus 3N.

I found that the first term was seven and I'm using the constant additive difference of negative three to plot the rest are my terms. The points align.

I know I'm right and then I erase the line.

Question three, two errors and a sentence to explain each.

You should have spotted the first term this sequence when N equals one, not when N equals zero, so that point should not have been plotted.

The term negative two does not align, so it must not be in the sequence.

Onto the second half of our learning now, where we're going to look at deducing the Nth term from a graph, we can look at an arithmetic sequence and deduce the Nth term when we see the sequence numerically.

In this case we look constant difference of positive four, so we know it's a 4N sequence, but it's not the sequence 4N.

It's a shift from the sequence 4N of positive one.

We could say that our sequence five, nine, 13, it's a translation of positive one from the sequence 4N, so it will come as no surprise to you that we call it four N plus one.

The four, the coefficient of N is the common difference in this arithmetic sequence.

The constant positive one is our translation.

It's also possible to look at the plotting of an arithmetic sequence and deduce an term rule.

We don't need to see the sequence numerically.

Generating the terms of 3N plus two reveals something about its structure.

We know the first term is five when N equals one, T equals five, but how is that term constructed? It's made of one lot of three plus the constant two.

What's that look like? You could say it looks like that.

What about the second term? That's eight, but what's it constructed of? It's two lots of three plus the constant two.

What will that look like? It'll look like that.

A third term, funnily enough, it's three lots of three plus the constant two and I think you know what it's gonna look like.

Well predicted, it looks like that.

What's coming next? The fourth term, what are you going to say about it? Well done, it's four lots of three plus the constant two and it looks like that, this is a visual representation of 3N plus two.

In each term we've got a constant of two and a varying number of threes.

Hence it's the sequence 3N plus two.

If you've got that, then you can get this, which arithmetic sequence is represented by this structure? Pause, tell the person next to you or say it to yourself.

See you in a moment to check your answer.

Welcome back.

I hope you said it's B, the sequence 2N plus three.

Why would you say that? Well, you'd say it because you can see in the structure of this sequence there's a constant of three in each term and a varying number of twos.

Hence the sequence is 2N plus three.

To get from the graph to the structure and therefore the Nth term rule, the common difference is a good starting point.

I wonder what this sequence is.

Let's pick out the common difference.

It's positive four.

Once we know that, we know we've got a varying number of fours in this sequence, we can represent those varying number of fours.

The first term's gonna have one lot of four.

The second term, two lots of four, the third term, three lots of four.

It's a 4N sequence, but it's not the sequence 4N.

We need to find that translation, that shift.

We can now see the constant in the translation from the horizontal axis.

Can you see what I mean by translation and shift? The N term rule is 4N plus three, a varying number of fours and a constant of three.

Quick check you've got that.

Let's find this Nth term rule.

Let's start with the common difference.

What's the common difference in this sequence? Pause and pick it out.

Welcome back, well done.

It is indeed positive five.

You can see that between the terms. Once we know that we've got a varying number of fives in this sequence, the next thing we need is the translation.

The shift from the horizontal axis two are varying fives, pause and pick it out.

Well spotted, it's positive two and there is the constant two in each term.

Now that we've got all that information and we've got this visual structure, can you pick out the Nth term rule? Pause, see if you can figure it out.

Welcome back, well done.

It is indeed option A, five N plus two, why? It's because there's a constant of two in each term and a varying number of fives.

We've got multiple methods for finding the Nth term rule from a graph.

We've already seen this graph.

We already know this one.

It's the sequence four N plus three, but I'm using it again to show you a different method for how we could have deduced this Nth term rule.

We know there's a common difference of positive four, so we know it's a 4N sequence, but what you don't yet know is if we draw a line through the terms, the Y intercept tells us the constant in our Nth term rule.

Look at the Y intercept, positive three.

Look at our Nth term, 4N plus three, wonderful.

This is the sequence 4N plus three.

It's a constant of three with a varying number of four.

Let's check you've got that.

What's the constant difference in this arithmetic sequence? You're good at this, it won't take you long Pause, see you in a second.

Well spotted, it's negative two.

You can see that between each term.

If we're gonna do this Nth term really quickly, then we need to know the Y coordinate at the Y intercept.

What is that Y coordinate the Y intercept when we draw a line through these points, pause and pick it out.

Welcome back, well spotted, it's option C, positive five.

Now that we've got all that information, can we pick out the Nth term rule? Pause, have a think about this one.

Welcome back and well said.

It's option C, five minus 2N.

Has a constant of five and a varying number of negative two.

The most efficient method of all needs even fewer visual aids.

Again, I'm gonna show you this same sequence for N plus three.

I'm gonna show you a different way that we could have deduced that Nth term rule.

We're interested first in the common difference, positive four, but from here, we're gonna select any coordinate pair and substitute into the general form for an Nth term rule of an arithmetic sequence, that general form being A N plus B, so a coordinate pair we could have chosen would be the first term, seven, when N equals one, T equals seven.

Let's substitute that into this general form.

We know something about this sequence.

It's a 4N sequence, but what's that constant? When we substitute in N equals one and T equals seven, we find that B, the constant is three, therefore the must be 4N plus three.

Check you've got that.

I'd like you to find the common difference, then select any coordinate pair and you substitution into the general form AN plus B.

To find the end term rule of this arithmetic sequence, Pause, give it a go.

Welcome back, the common difference positive six.

Any coordinate pair we can choose to substitute in.

You didn't have to choose the same one as me.

When I substitute this in, I found a constant to be negative five.

Therefore the rule is 6N minus five.

Whether you chose that first coordinate pair, second coordinate pair or you chose a third coordinate pair like me, you should have found the constant in our Nth term rule to be negative five and the Nth term rule to be 6N minus five.

We only need a few points of a sequence to solve some very difficult problems. Here on the graph, we can only see three terms of this arithmetic sequence, but that's all we need to answer.

Two questions like, is 295 in the sequence and what is the first term greater than 1000 in the sequence? Obviously I'm not going to grow that graph to find if 295 is in the sequence.

I'm certainly not gonna continue it to see if I can find the first term greater than 1000.

That would take me a lot of time and a lot of space.

So a more efficient method is to take those three terms that we know and use them to deduce an N term.

We know the Nth term is 6N minus five.

We can use that to see if 295 is in the sequence.

In order to do that, we form this equation.

Does our sequence 6N minus five hit 295? Let's have a look.

If 6N minus five equals 295, 6N equals 300 and N equals 50, wonderful.

295 is in the sequence, in fact, it's the 50th term.

Next problem, what's the first term greater than 1000? I don't know, but let's see.

When does our sequence hit 1000? It hits 1000 when N equals 167.

5, but we know there's no 167.

5th term, so we'll find the 167th term and the 168th term and that's all the ammunition we need to be able to say, the 168th term is the first term greater than 1000.

Quick check you've got that.

What's the first term smaller than negative 200 in this sequence, pause.

Give this one a go.

Welcome back.

Hopefully you started by spotting the common difference, negative five.

We can use the intercept method to deduce an N term rule, 15 minus 5N.

From there we want to know when that sequence hits negative 200.

It hits negative 200 when N equals 43, so the 43rd term is negative 200.

What's the 44th term? Negative 205, what's our conclusion? The 44th term, negative 205, is the first term smaller than negative 200.

Practise time now, question one, part A, I'd like you to deduce the N term of this arithmetic sequence.

For part B, I'd like at least one sentence explaining how you know that is the N term.

Pause and do this now.

Question two, I'd like you to deduce the Nth term of these two arithmetic sequences, but I'd like you to use a different method for each.

So whatever method you use for A, you're not allowed to use for B.

You've gotta do something different.

This is a good opportunity for you to practise your mathematical fluency.

That's a good thing.

Pause and do this now.

Question three, we can only see one small region of the graph of this arithmetic sequence.

Here's the first three terms. The question is what is the last positive term in the sequence? You'll enjoy this one.

Pause, good luck.

Feedback time, let's see how we did.

Part A, should have said the Nth term rule is 7N plus three, at least one sentence explaining how we know that, you might have written, the structure shows us there is a constant of three and a varying number of sevens, hence the rule is 7N plus three.

For question two part A, I'm looking at the common difference and then drawing a line through the sequence to find that interceptor positive one.

So I can declare the Nth term rule is 3N plus one.

For B, I need to use a different method, so I'm finding the common difference and I'm using a coordinate pair substituting into the general form, A N plus B, and I find that the rule is 5N minus four.

Question three, we're looking for the last positive term in this sequence.

We need the Nth term rule for the sequence.

I can do that by finding the common difference, substituting a coordinate pair into the general form.

The Nth term rule is 215 minus seven N.

What's the last positive term? IE, when does this sequence reach zero? Well, the sequence reaches zero when N equals 30.

71 and so on, the 30th term is positive five.

The 31st term is negative two, so I can conclude the 30th term, five is the last positive term.

We're at the end of the lesson now sadly, but we have learned that an arithmetic sequence can be shown graphically by plotting coordinate pairs from a table of values or by plotting points using the common difference.

We've also learned that when we see an arithmetic sequence graphically, we can relate it back to its Nth term formula from its structure or by identifying the common difference and intercept or by substituting into the general form A N plus B.

I hope you enjoyed this lesson as much as I did and I look forward to seeing you again soon for more mathematics.

Goodbye for now.