Lesson video

Hello, Mr. Robson here.

Welcome to maths.

What a lovely place to be representing geometric sequences graphically today.

Geometric sequences, graphs, what's not to love? Let's find out what it's all about.

Our learning outcome is that we'll be able to represent geometric sequences graphically and appreciate their common structure.

Keywords that you'll hear throughout this lesson, geometric sequence.

I'll also use the phrase common ratio a lot.

And we're gonna start by graphing a common ratio greater than one.

Andeep and Sam are making models of sequences.

Andeep says, "My sequence is arithmetic.

It starts on one and has a term-to-term rule of plus two." Sam says, "My sequence is geometric.

It starts on one and has a term-to-term rule of multiply by two." I wonder if they'll be different.

Let's have a look at Andeep's.

It starts on one, adds two, adds two more, adds two more and adds two more.

That's a model of the arithmetic sequence starting on one with a term-to-term rule of add two.

Sam says, "I can see why they are also called linear sequences.

It's growing in a straight line." Let's have a look at Sam's model now.

Remember Sam said "My sequence is geometric.

It starts on one and has a term-to-term rule of multiply by two." Andeep rightly ponders, "I wonder if your sequence will look different." Let's see, shall we? There's the first one, and then let's multiply that by two.

Let's multiply that by two.

Let's multiply that by two, and multiply again by two.

"That's no straight line," says Andeep.

Well done Andeep.

This is not a linear sequence, something different's happening.

If we look at how these sequences are building, in Andeep's sequence, we were adding two, and adding two, and adding two, and adding two.

A constant additive difference in the arithmetic sequence is what gave us our linear increase.

Contrast that with Sam's model of a geometric sequence, we were doubling one, and then doubling two, and then doubling four, and then doubling eight and so on.

This geometric sequence has an increasing increase, therefore it forms a curve.

Andeep says, "Your geometric sequence had a common ratio of multiply by two.

What would a common ratio of multiply by three look like?" Sam says, "I don't know, let's explore it." And that's the wonderful thing about mathematics, a lot of it is exploration.

So the geometric sequence starting on one with a common ratio of two, we've seen that modelled already.

It went 1, 2, 4, 8, 16.

So how will it look different when we have a geometric sequence starting on one with a common ratio of three? The first term, the second term, the third term, the fourth term, and the fifth term doesn't even fit on the screen.

We went from one, treble that, treble that, treble that, we're quickly at 27, and 81, the next term will be miles off the top of this screen.

The same curved shape but with a higher common ratio, giving us an even steeper curve.

We see this same curved shape when we graph geometric sequences.

You may have seen graphing of sequences before, where we make the horizontal axis, the term number N, and the vertical axis, the term value T.

If I said let's plot a geometric sequence with a first term of five and a common ratio of two, you'd be plotting the terms 5, 10, 20, 40, 80.

We turn those into coordinates, 1-5, 2-10, 3-20, 4-40, 5-80, and when plotted, they look like that on a graph.

If I add a line there, it demonstrates the curved shape of increasing gradient more clearly.

However, we don't usually draw the line when plotting sequences because there's no 2.

5th term.

I will show you the line again though because that enables us to see something about the nature of the sequence.

Increasing the common ratio increases the speed of the rate of change.

What do I mean by that? The graph you can see there is a first term of five and a common ratio of two.

What if I said let's have a first term of five and a common ratio of five? You'd get these terms, 5, 25, 125, 625, 3,125.

When we go to plot that, it very quickly accelerates off our axis here.

You can see the difference in steepness between our common ratio two curve and our common ratio five curve.

Because the rate of change with a common ratio of five is so fast, we need to adjust the scale on the axis.

When I do that, you can see the common ratio five making that same upward curve, but you can also see how quickly it's accelerating away.

A quick check you've got that now.

Which of these graphs could show a geometric sequence, is it graph A, graph B or graph C? One point to note, the scales for T and N increase in equal amounts for each of those graphs.

So, which one is geometric A, B, or C? Pause now, have a think.

Have a conversation with the person next to you.

I'll see you in a moment for the answers.

Welcome back.

I hope you said it's not A.

A's linear, that must be an arithmetic sequence.

Did you say something similar for B? Well done.

It's linear.

That must be an arithmetic sequence, whereas C could be our geometric sequence.

We see an increasing upward curve, that might be a geometric sequence.

A common ratio of multiplier by five gave us a rapidly increasing curve, like the one you see in that graph there.

Andeep is considering another ratio.

What if the common ratio was just 1.

2? That's really small by comparison to five.

Would that still form a curve? Let's have a look, shall we? Common ratio of 1.

2, we could start with the first term of 10,000, have a common ratio of 1.

2 and we'd get these terms. If we plotted them, the graph would look like this.

Andeep says, "It looks almost linear," and I agree, it does look almost linear, but it's not quite linear, is it? So what's going on here? Continuing the sequence to the 10th term, and you'll be able to see that despite this being a smaller multiplier, we still get that same increasing upward curve.

If I show you the sixth term, the seventh term, the eighth term, the ninth term, the 10th term, can you see the graph now is really starting to take off? It's that same beautiful upwardly curved shape.

This graph is a great example of where you'll see this maths in the real world.

Without changing any of those coordinates, I can change the context to a real life example.

I could say a business invests 10,000 pounds in shares of a green energy company.

Their value grows by 20% every year.

Instead of talking about plotting N, the term number, versus T, the term value, we're now talking about plotting T, time, versus pounds, monetary value of those shares.

We'd still get the same values in the table because we're still at the same common multiplier of 1.

2.

If you grow by 20%, that's a multiplier of 1.

2.

It'll give us the same terms. But by modelling this on a real life context, you see the longer they leave their investment, the faster it's growing.

Quick check.

You've got that concept of how quickly geometric sequences can grow.

Here are the graphs of three geometric sequences with lines drawn to show their upward curves.

Match the common ratios to their respective line.

There's three lines on the graph, one of them is representing a common ratio of two, one of them is representing a common ratio of 1.

5, and one of them is representing a common ratio of four.

But which is which? Pause, have a conversation with the person next to you, or a good think to yourself, and I'll see you in a moment for the answers.

Welcome back.

Did we say the middle line must be the common ratio of two? Because that's in the middle of 1.

5, 2 and 4.

Did we say the lowest line with the slowest increase is the lowest common ratio, 1.

5? Did we say the most quickly increasing line is the common ratio of four? If so, well spotted.

Practise time now.

Question one.

I'd like you to fill in the table of values and plot the geometric sequence that starts on one and has a common ratio of three.

Pause, fill in the table, plot those points.

I'll see you in a moment.

Question two, you invest 2000 pound in a new tech company and your shares grow in value by 50% each year.

Calculate their value for the first eight years and plot it on the graph.

Couple of hints for you.

A 50% increase is a multiplier of 1.

5.

Use that between successive terms. When calculating those terms, it'll be quicker and more efficient if you just round each value to the nearest pound, and then when plotting those on the graph, you won't be able to do it to a high level of accuracy.

You'll have to approximate where those points are, but you'll still see the nature of this sequence.

Pause, give this problem a go now.

Question three, Sam is contemplating the shape of the graphs of geometric sequences.

Sam says, "I don't know if the starting term affects the shape.

If I use the same common ratio of two but start on two different numbers, will I get a different shape?" You'll notice on the left hand table a first term of two, on the right hand table, a first term of eight.

I'd like you to use a common ratio of two to complete both of those tables, then plot both graphs and write a conclusion for Sam.

Feedback time.

Question one.

We had a geometric sequence starting on one with a common ratio of three.

Your terms would've been 1, 3, 9, 27, 81.

You're plotting the coordinate 1-1, 2-3, 3-9, 4-27, 5-81, and your sequence would look something like that when graphed.

You might wanna pause now and just check you've got the same terms as me with coordinates in similar positions.

For question two, we've invested 2000 pound in a new tech company.

Their shares are growing by 50% each year.

That's a common ratio of 1.

5.

Generating these values.

We'll go from 2000 pound, to 3000 pound, to 4500 pound, to 6,750, et cetera.

Remember I told you to round each value to the nearest pound.

By the time you got to the eighth year, you should have had 34,172 pounds.

When we plot those coordinates, the coordinate 1-2000, 2-300, 3-4500, we get this shape.

The points plot the curved pattern of a geometric sequence.

Look at how quickly your investment is growing in those last couple of years.

By the way, if ever you do come across any companies who share values are going up by 50% year on year, do let me know about them.

Question three, Sam's contemplating the shape of graphs of geometric sequences.

Using the same common ratio of two but starting on two different numbers, will Sam get a different shape? Starting on two with a common ratio of two, we get the terms 2, 4, 8, 16, 32.

Starting on eight with a common ratio of two, we get the terms 8, 16, 32, 64, 128, and then when you plot them on the graph, they look like that.

I will draw lines just to show you the difference between the common ratio between starting on two with a common ratio of two, between starting on eight with a common ratio of two, and then you can conclude both graphs have the same shape.

They both represent an upward curve getting increasingly more steep, even though we had a different first term.

Onto the second part of our lesson now, graphing a common ratio ratio less than one.

Izzy and Jacob are plotting geometric graphs.

You've already seen this graph, a first term of 10,000, a common ratio 1.

2.

Jacob says, "Geometric graphs are so pretty." I'm inclined to agree, Jacob.

"They always make an upward curve, no matter how small the common ratio." Izzy says, "Are you sure, Jacob? Have you tried a common ratio of one?" I don't think we have yet this lesson, so let's have a look at what that looks like.

If we start on 10,000, have a common ratio of one, we get the terms 10,000, 10,000, 10,000, 10,000, 10,000 and it plots like so.

Jacob says, "Of course, the ratio is always one to one.

The value never changes." Izzy says, "Yes, you get a straight horizontal line from your first value." That is what a common ratio of one looks like.

Let's keep going with this exploration.

We've seen in this lesson what a common ratio of two looks like.

We now know what a common ratio of one looks like.

Jacob very sensibly asks, "So what shape would a common ratio of a half give us?" Great question, Jacob.

Izzy's got the right attitude.

"I don't know.

Let's find out." Let's make a table of values.

We can start on that first term of 10,000 and apply a common ratio of a half.

We'll get the terms, 10,000, 5,000, 2,500, 1,250, 625.

When we plot those, the graph looks like this.

Jacob says, "Of course, the values decrease." Izzy says, "It still makes a curve but it's a downward curve." Great observations, Jacob and Izzy.

Well done.

The line shows us the nature of the curve.

It's decreasing but at a rate which is slowing.

However, we don't draw the line for this sequence.

There is no one point fifth term.

"So will we get this shape for all ratios less than one?" asks Jacob.

Good idea.

Let's keep exploring.

Izzy suggests, "I don't know.

Let's try one fifth." We can start with a first term of 10,000, use a common ratio of a fifth and we'll get these terms, 10,000, 2,000, 400, 80, 16.

When plotted, it looks like that.

If I draw a line to show you the nature of this, you can see it's the same shape, just a much steeper curve.

"What an interesting graph," says Izzy.

I do hope you agree.

Quick check you've got this now.

Which of these graphs shows a geometric sequence with a common ratio of one third? Is it A, is it B, is it C? And be aware the scales for T and N increase in equal amounts on all these graphs.

So common ratio of a third, is that graph A, graph B or graph C? Pause, have a conversation with the person next to you or a good think to yourself.

See you in a moment with the answers.

Welcome back.

I hope you said it's not A.

Why is it not A? Because that's like the geometric sequences we saw in the first part of the lesson, where the common ratio was greater than one.

They were increasing upwards.

I hope you said it's not B.

We know that's linear.

That's an arithmetic sequence, not a geometric one.

It's C, a geometric sequence with a common ratio of one third would look like a decreasing curve.

Common ratio of one fifth looks like so.

Jacob says, "Wait, we didn't test all types of common ratios less than one." What do you think Jacob could be referring to? Pause this video, make a suggestion to the person next to you or aloud to me on screen.

What could Jacob mean? Welcome back.

Izzy's on Jacob's wavelength.

She says, "Do you mean negatives?" They're less than one.

What if we try a common ratio with a negative value? If we started at a first term of one and had a common ratio of negative two all the way along the sequence, we would get one, negative two, positive four, negative eight, positive 16, negative 32, positive 64, negative 128.

What an interesting sequence.

When we plot it, the graph looks very different indeed.

Jacob says, "What a strange graph." Izzy says, "It's oscillating between positive and negative terms." What a lovely bit of vocabulary, Izzy, oscillating.

It's bouncing from positive to negative to positive to negative.

You can see it oscillating if I draw a line between those points.

When the common ratio of a geometric sequence is negative, the terms oscillate from positive to negative.

You might notice if I draw a line through the positive terms and then draw a second line through the negative terms, you might notice that the positive terms still form a curve, and the negative terms form their own curve.

Quick check you've got that.

This is the graph of a geometric sequence.

What could the common ratio be? Is it A, five, B, a fifth, or C, negative five? Pause, have a conversation with the person next to you.

I'll see you in a moment for the answers.

Welcome back.

I hope you said it's not A, it's not B, it's C, negative five.

When the terms oscillate from positive to negative, the common ratio must be negative.

Practise time now.

Question one, part A, I'd like you to complete the table of values and graph this geometric sequence with a common ratio one third.

Plot the coordinates as accurately as you can.

You'll only be able to approximate their position, but you'll still see the nature of this sequence.

Pause.

Give this problem a go now.

Question 1B, looks incredibly similar to 1A, but there's a crucial difference.

Complete the table of values and graph this geometric sequence with a common ratio of two thirds.

Can you fill in that table and plot the coordinates as accurately as you can? I appreciate you can only approximate them on this grid.

Pause.

Give this a go now.

Question two.

Complete the table of values and graph this geometric sequence with a common ratio of negative 1.

5.

Pause and do that now.

Feedback time now.

Table of values with a common ratio of one third and then the graph of that geometric sequence.

The values would've been 2,025, 675, 225, 75 and 25, and the graph would look like so.

Common ratio between zero and one.

It gives us a downward curve.

For question one part B, common ratio of two thirds this time, giving us the terms 2,025, 1,350, 900, 600, 400.

Your graph looks like so.

Again, a common ratio between zero and one giving us a downward curve.

The difference this time being it's slightly less steep than the one third curve you saw in question 1A.

Question two is the one where we had a negative common ratio.

Common ratio of negative 1.

5 across the table of values would give us these terms. When we plot those, we get a graph like so.

Negative common ratio giving us oscillating terms from positive to negative.

You can see that the positive terms form a curve and the negative terms form a curve.

That's the end of lesson now.

In summary, we can represent geometric sequences graphically and appreciate that some of them form an upward curve, some of them form a downward curve, and some oscillate from positive to negative.

Hope you've enjoyed this lesson as much as I have, and I look forward to seeing you again soon from more mathematics.

Goodbye for now.