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Hello, Mr. Robson here.
Welcome to Maths, it's a great place to be.
Today we're recognising geometric sequences.
You gotta love a geometric sequence, so let's get going.
Our learning outcome is I'll be able to recognise a geometric sequence.
Keywords that you'll hear throughout this lesson, geometric sequence, of course.
A geometric sequence is a sequence with a constant multiplicative relationship between successive terms. Common ratio, a common ratio is a key feature of a geometric sequence.
It's the constant multiplier between successive terms, this is called the common ratio.
You will see that a lot throughout this lesson.
Two parts to our lesson today, and we're gonna begin by finding the common ratio.
Sofia and Aisha are looking at some sequences.
One of the sequences goes seven, 16, 25, 34, 43, the other nine, 15, 22, 30, 39.
Sofia says, I think these sequences are arithmetic, but I can't remember how to check.
Luckily, Aisha's on hand to help, and it's lovely to see pupils helping each other out in the classroom.
Aisha says, take each term and subtract the previous term.
And if you find a common additive difference between each term, the sequence is arithmetic.
Sofia tests this.
43 subtract the previous term of 34 gave her nine.
34 minus the previous term of 25, nine again.
25 minus the previous term, 16, gives us nine again.
And 16 minus seven gives us nine again.
So, you've got a common additive difference of positive nine between the terms of that sequence.
Aisha says, a common additive difference, this sequence is arithmetic.
So that one's arithmetic, when we look at the one on the right hand side, however, something's different.
39 minus the previous term of 30 gives us nine.
30 minus the previous term of 22 gives us eight.
22 minus 15 gives us seven, and 15 minus nine gives us six.
So the sequence adds six, then it adds seven, then it adds eight, then it adds nine.
It's not a common additive difference.
There is a pattern, but it is not arithmetic as a sequence.
Thank you, Aisha.
Sofia then says, is there a similar test I can apply to identify if a sequence is geometric? Geometric, there's a subtle difference between testing for a geometric and testing for an arithmetic sequence.
Aisha again is on hand to help us.
Yes, take each term and divided it by the previous term.
If you find a common multiplicative relationship between each term, the sequence is geometric.
So Sofia tests it.
162 divided by 54, three.
54 divided by the previous term, 18, that's three.
18 divided by six, three.
Six divided by two, three.
A common multiplicative relationship between consecutive terms, this sequence is geometric.
Well done, Sofia.
The common multiplier is called the common ratio.
For this sequence, I could say, it's a common multiplier of three.
I could say, it's a common ratio of three.
This sequence has the first term, two, and the common ratio, three, and this uniquely defines this geometric sequence.
You can also see why it's called a common ratio.
Each pair of terms is in the exact same ratio.
You might have seen these called equivalent ratios before.
How do I know that they're equivalent ratios? Well, if I cancel each ratio to its simplest form, two to six cancels down to one to three, six to 18 cancels down to one to three.
you'll never guess what 18 to 54 cancels down to.
Well done, one to three.
54 to 162, it takes a bit more cancelling down, but it's still in the ratio of one to three.
That's why you'll see it called the common ratio, but we don't tend to call it one to three, we tend to just say that's a common ratio of three.
Quick check that you've got what I've said so far.
To find the common ratio in a geometric sequence, you should, is it, A, multiply consecutive terms together? Is it B, subtract the previous term from each term? Or is it C, divide each term by the previous term? Pause this video, tell the person next to you, and I'll see you again in a moment.
Welcome back.
What did we go for? I hope we didn't go for A, because we don't multiply consecutive terms together.
I hope we didn't go for B, subtracting the previous term from each term, that's how we search for an arithmetic sequence.
We should have gone for option C, it's a geometric sequence we're looking for, dividing each term by the previous term.
Another quick check, find the common ratio in this geometric sequence.
You're welcome to use a calculator, I certainly would for some of these divisions.
Pause, find the common ratio.
I'll see you again in a moment.
Welcome back.
How do we get on? Did we do 322,102 divided by 29,282, and get 11? Did we then divide 29,282 by 2,662 and, again, get 11? And the pattern continues, when we divide the third term by the second term, we get 11.
When we divide the second term by the first term, we get 11.
By dividing each term by the previous term, we find a common ratio of 11.
I could write that underneath our sequence like that.
Common multiplicative relationship, times 11, between each successive terms, that's the common ratio, 11, at the heart of our geometric sequence, Sofia and Aisha are looking at another sequence.
80, 200, 500, 1,250, 3,125.
Sofia says, this sequence can't be geometric because 80 is not a factor of 200, and 200 is not a factor of 500.
80 is not a factor of 200, she's right.
80 times two is 160, 80 times three is 240.
80 is not a factor of 200.
So Sofia concludes that this means it can't be a geometric sequence.
Aisha, again, on hand to help.
Test it, take each term and divide it by the previous term.
If you find a common multiplicative relationship between each term, this sequence is geometric.
So let's test this sequence.
200 divided by 80 is 2.
5.
500 divided by 200 is 2.
5.
1,250 divided by 500, 2.
5.
You'll never guess what's coming next.
3,125 divided by 1,250, and it is 2.
5.
So, we've got a common multiplicative relationship of times 2.
5 between successive terms. So Sofia says, this sequence is geometric, it has a decimal common ratio.
The common ratio does not have to be a whole number, it does not have to be an integer, it could be a decimal.
But Aisha suggests it's better to think of it as a fractional multiplier, rather than calling the common ratio 2.
5, in this case, 2.
5 is a fraction.
Well, that's two and a half, which is five halves.
We'd write 5/2 as a fraction for 2.
5, and Aisha says it's better to think of it as a fractional multiplier.
So, where did that fractional multiplier come from? Rather than typing 200 divided by 80 into your calculator to get 2.
5, you could consider it as 200/80.
It's one and the same thing.
When we cancel that fraction down, it cancels to 5/2.
500 divided by 200, we could call that 500/200, cancel out a fraction down, 5/2.
1,250/500 cancels down to 5/2, as does 3,125/1,250.
So, we could say we've got a common ratio of 5/2.
Quick check you've got that.
Lucas is trying to find the fifth term of this geometric sequence.
216 divided by 162 is 1.
33 to two decimal places.
288 divided by 216 is 1.
33 to two decimal places.
384 divided by 288 is 1.
33 to two decimal places.
It looks like we found a common ratio.
So, Lucas says the common ratio is 1.
33.
So the next term is 284 multiplied by 1.
33, which gives us 510.
72.
What advice do you think Aisha is going to give Lucas, and why? Tricky question this, so you'll want to pause, have a good think and maybe a conversation with the people around you, and I'll see you again in a moment.
Welcome back.
Aisha's advice to Lucas is, leave your common ratio in fraction form for accuracy and efficiency.
Your calculator will help you to do this.
If you'd found the common ratio by doing 216 divided by 162, your calculator would've told you it's 4/3.
There's no need to convert that to a decimal.
As we divide any term by the previous term, we continue to get the result of 4/3.
288 divided by 16, 4/3.
384 divided by 288, 4/3.
Leave the common ratio as 4/3.
Don't convert it to a decimal, because then we have a common ratio of 4/3.
When we find the next term, 384 multiplied by 4/3, we get 512, which is the actual fifth term.
Thanks, Aisha, says Lucas.
You're welcome, Lucas is I'm sure what Aisha would say back.
Lucas and Aisha find this sequence, five, 10, 30, 120, 600.
I'm enjoying geometric sequences.
This one has a really strange pattern, says Lucas.
Aisha intervenes with, remember, it has to be a common multiplicative relationship between each term in order for the sequence to be geometric.
The multiplier changes in your sequence, so it's not geometric.
Just because there's a multiplicative relationship between the terms here doesn't make it geometric, it has to be a common multiplicative relationship.
A geometric sequence must have a constant common multiplicative relationship between terms. Lucas's sequence here is not geometric.
Watch out for those.
Let's check you've got that.
Which of these are geometric sequences? Use your calculators, I'll see you again in a moment.
Welcome back.
Let's see how we did.
We should have spotted a common multiplicative relationship of times three between successive terms. In the sequence in A, we could say it's a geometric sequence.
For B, however, we had a multiplier of two from the first term to the second term, and a multiplier of three from the second term to the third term, and then it goes times two, times three.
It's a lovely pattern, but it's not a constant multiplier, therefore it's not geometric.
For C, a useful pattern this.
Multiply by two in the first case, multiply by four, then multiply by eight, then multiply by 16 to get from term to term that sequence.
It's a beautiful pattern, but it's not a constant multiplier.
Therefore, that is not a geometric sequence.
How about D? I hope you spotted a constant multiplicative relationship of times five between successive terms, therefore said, that one is geometric.
Well done.
Practise time now, question one.
I'd like you to divide each term by the previous term to find the common ratio in each of these geometric sequences.
You're well practised at these now, so I'll leave you to it.
Pause, and I'll see you again in a moment.
For question two, part A, this sequence is geometric.
I'd like you to show that it is geometric, and write a sentence to justify why it is.
For part B, I've given you a sequence which is not geometric.
I'd like you to demonstrate that it's not geometric, and I'd like you to write a sentence to justify why it is not.
Pause and do that now.
Feedback time.
Dividing each term by the previous term to find the common ratio of geometric sequences, we should have found the fifth term divided by the fourth term gave us seven.
The fourth term divided by the third term gave us seven, and so on and so forth.
We found a common ratio of seven.
For B, fifth term divided by the fourth term gives us nine.
Fourth term divided by the third term gives us nine, and so on and so fourth.
We have a common ratio of nine for the sequence in B.
C, slightly different, but no different in reality.
A term divided by the previous term gave us 1/3, a term divided by the previous term gave us 1/3.
63 divided by 189, 1/3, 189 divided by 567 gives us 1/3.
Your calculator should tell you that answer as a fraction, and you should leave it as a fraction, and say that one has a common ratio of 1/3.
For D, we had to divide a fraction by a fraction, but as we divided any term by the previous term, we continue to get a common ratio of 3/2.
Onto question two, I asked you to justify why this is a geometric sequence.
You should have shown that it's a geometric by dividing each term by the previous term, and revealing a common ratio.
That's how you would show somebody that this is a geometric sequence.
You would then write something along the lines of, the sequence has a common ratio of 3/4 between successive terms, therefore is geometric.
You could have justified it another way.
You could have looked at the sequence going from 768 to 576, from 576 to 432, and so on, and written those as ratios.
If you then cancel down those ratios, you would find constantly, they cancel in their simplest form to 4 to 3.
So you might have said, the sequence has a constant ratio of 4 to 3 between successive terms, therefore is geometric.
But do remember that the common ratio should be given as a single value.
So you would still say this is a common ratio of 3/4, the multiplicative relationship from four to three.
For part B, I asked you to show that this sequence is not geometric, and write a sentence to justify why.
As you divide each term by its previous term, we get different results.
That will be enough to show that this is not a geometric sequence.
From the first term to the second term, a multiplier of 1/3.
From the second term to the third term, a multiplier of 1/2.
And then multiply by 1/3, multiply by 1/2, it's a lovely pattern, but it's not constant.
It's not a common multiplier, so it's not geometric.
So you should have written something along the lines of, this sequence has a multiplicative relationship of 1/3 between some terms and 1/2 between others.
Therefore, it is not geometric.
You could also have said that when cancelling those ratios down, you get a variety of results.
The sequence has a ratio of three to one between some terms, and two to one between others.
That's not consistent, therefore it's not geometric.
That'd be another way to justify why that's not a geometric sequence.
Onto the second half of the lesson now.
We're gonna be using the common ratio.
Sam and Jacob are looking at missing terms in sequences.
We've got a blank in the first term space, a blank in the second term space.
We know the third term, we know the fourth term, and a blank for the fifth term.
Sam says, "This is an arithmetic sequence with a common difference of positive 21." "Can you find the missing terms?" Jacob says, "Yes, I just add 21 to find the fifth term, and then I do the inverse of negative 21 to find the previous terms." So, if as we increase in our sequence, we're going up by positive 21, the next term must be 126 plus 21, 147, and it is.
So what does Jake mean by the inverse? Ah, if we wanna move backwards, back to the second term and the first term, we have to subtract 21 rather than add it.
So 105 minus 21 gives us 84, 84 minus 21 gives us 63.
Does that look right? Sam, you're one step ahead of us.
Thank you.
I've checked it, you're right, Jacob.
There's a arithmetic sequence with a common additive difference of 21 between successive terms. Sam and Jacob are still looking at missing terms, a different sequence this time.
And we have the third term and the fourth term, but we're missing the first one, the second one, and the fifth one.
Sam says, oh, exciting, this is a geometric sequence with a common ratio of five.
Can you find the missing terms of this one? Jacob can, and he applies a pretty similar principle, but with a important difference.
Yes, I just multiply by five to find the fifth term, and then I do the inverse.
What's the inverse of multiply by five? Yes, divide five, to find the previous terms. So moving forwards in the sequence, and multiplying by five to get from term to term.
So the next term, the sequence, the fifth one must be 75 times five, it must be 375.
So to move backwards through our sequence, we do the inverse, which is to divide by five.
15 divided by five is three.
What's coming next? Three divided by five.
That's three, divided by five, which we can just write as 3/5.
Remember, communicating that as a fraction, really quick and efficient to do so.
Once again, Sam says, I've checked it and you're right, Jacob.
There we are, our geometric sequence with a common ratio of five.
Quick check you've got that now.
What's the fifth term of this geometric sequence? The fifth term, feel free to use your calculator.
Pause this video, see if you can work it out.
I'll reveal the answer in a moment.
So, first step, I hope you took 968 divided by 88 is 11.
That's our common ratio between successive terms, 11.
So to find that fifth term, we have to do the fourth term, 968 multiply by our common ratio, 11, to get 10,648.
Next check, what is the second term of this geometric sequence? Keep your calculated to hand, see if you can work that out.
Pause this video, I'll reveal the answer in a moment.
Welcome back.
How did we get along? If we're moving backwards in this sequence, we do the inverse of multiply by 11, that is to divide by 11.
So we need to do 88 divided by 11 and we get eight.
Did you get that? Well done.
Finally, same sequence, but what is the first term? I've given you some options this time, because there's a common misconception that I'm looking to talk about.
Let's see if you can spot the misconception, as well as spotting the right answer.
What's the first term of this geometric sequence? Is it 88? Is it 0.
727? Or is it 8/11? Pause, have a good think, have a conversation with the people around you.
I'll see you in a moment.
Welcome back.
I do hope you quickly ruled out option A, it's not 88.
To find a previous term, we need to do the inverse of times 11, that would be to divide by 11 rather than taking eight and multiplying it by 11.
How about option B, 0.
727? I bet some people went for that, but it's not the right answer.
If you typed in eight divided by 11 into your calculator, you were on the right path, but when your calculator told you the answer is 0.
727272 with a 72 recurring, you need to leave it with 100% accuracy.
As soon as you round to something like 0.
727, you lose accuracy.
So, the answer was C.
Eight divided by 11, we just write as 8/11, and that's the most accurate and concise way to communicate that number as our first term.
It's also possible to find missing terms in geometric sequences when you aren't given successive terms. We're not given successive terms, i.
e.
, we don't know the first one, we do know the second one, we don't know the third one, we do know the fourth one.
There's a gap between the two terms that we know.
So can we find the missing terms here? Well, Jun's on it.
Eight divided by two is four.
So the common ratio must be four.
Hmm.
Jun's not right.
How do you know? Pause, have a think, have a conversation with the person next to you.
How do you know that the common ratio's not four, Jun's not right? See you in a moment.
Welcome back.
How do we know that Jun is not right? Well, if it was a common ratio of four, then the third term would be the second term, two multiplied by four, it would be eight.
That would make the fourth term eight multiplied by four, which is 32, but we know that's not the truth.
The fourth term is eight, it is not 32.
So we know it's not as simple as just dividing eight by two, i.
e.
, the two terms we knew.
Something else is going on here.
Jun says, I checked, the common ratio can't be four.
Well done, Jun, it's important to check your work in mathematics.
It must be something that multiplies by itself twice to make four.
So the common ratio, we'll call it R for a moment, two multiplied by R will give us the third term, and then multiply by R again to give us the fourth term.
We could set this up as an equation.
Two multiplied by the common ratio twice to get to eight.
We can simplify that, two lots of R squared to get eight.
And then we could divide both sides of that equation by two.
So R squared must equal four.
Jun's happy now, it's two.
When you square two, you get four.
And that's absolutely right.
The ratio must be two.
If we show the common ratio of two between successive terms, we'll find a fifth term of 16, a third term of four, and a first term of one.
Well done.
June says, I'm so proud of myself, and he should be.
But Izzy's on hand to say, don't celebrate too soon, there are actually two solutions to this problem.
So, two multiplied by R twice to get to eight, R squared equals four, that's where Jun got to.
And then he rightly asked himself the question, something else which squares to make four.
Got it, negative two.
Well done, Jun.
Two squared makes four.
But also, negative two squared makes positive four.
So there were two solutions to this equation, positive two or negative two.
If we apply a common ratio of negative two, we would find a fifth term of negative 16, a third term of negative four, and a first term of negative one.
Well done, Jun, you found both possible sequences, says Izzy.
Let's check you've got that now.
What could the common ratio be for this geometric sequence? Notice, we know the second term of six, the fourth term of 54, but we don't know successive terms. There's a gap in the middle there.
So what could the common ratio be, three, negative three, or nine? Pause, have a think, have a calculate.
See if you can figure this one out, I'll see you in a moment.
Welcome back.
I hope you said the common ratio could be positive three.
If we had a common ratio of three between terms, we'd get a first term of two, and then it goes to six, a third term of 18, multiply that by three to get to 54 and then multiply that by three to get to the fifth term, 162.
That common ratio three works for this geometric sequence.
But there was a second solution, negative three also works.
If we apply a common ratio of negative three, the sequence would go negative two, positive six, negative 18, positive 54, negative 162.
Similar, but a crucial difference, I hope you see.
We should have ruled out option three.
Just because 54 divided by six is nine, doesn't make nine the common ratio.
Practise time now, question one, I'd like you to find the missing terms in these geometric sequences.
You'll need your calculator.
Pause, give these a go, I'll reveal the answers shortly.
Question two, find the two possible geometric sequences given these missing non-successive terms. We know the first term is four, the third term is 100.
There's two possible geometric sequences.
Can you find them? Pause, again, have a play with your calculator.
I'll see you shortly with the answer.
Feedback time, question one, finding the missing terms of these geometric sequences.
So for A, we know the fourth term to be 153, we know the third term to be 51.
That must be if it's a geometric sequence, a common ratio of three between the terms, which will give us the sequence, 17/3, 17, 51, 153, 459.
I do hope you communicated that first term as 17/3, as a fraction, rather than converting that to a decimal.
For B, common ratio was 45 divided by 225, giving us 1/5, a decreasing geometric sequence, common ratio, 1/5.
That gives us the missing terms 1,125, and then 225, 45, nine, and 9/5.
For C, 2.
8 divided by the previous term, 0.
14, gives us 20.
A common ratio of 20 between successive terms would give us a sequence.
0.
007, 0.
14, 2.
8, 56, 1,120.
Now, I know I said communicate as a fraction, but if we're given the question as a decimal, there may be a context where we use decimals in geometric sequences.
So I wrote the rest of my terms as decimals in that moment.
For D, we need to do 56/81 divided by 28/27 in our calculators to get 2/3.
That's the common ratio, 2/3, giving us the terms 7/3, 14/9, and the fifth term of 112/243.
You might wanna pause at this screen now, and just check that your working out your common ratios and your terms are the same as mine.
Onto question two, we don't know successive terms in this geometric sequence, that means there's two possible sequences.
We know that four multiplied by the common ratio twice gets us to 100.
We can simplify that equation to four multiplied by R squared.
Divide both sides by four, R squared is 25.
Something squared is 25.
There's one obvious answer, five, and the less obvious answer, negative five.
Negative five squared is positive 25.
So we could add a common ratio of positive five, which would've given us a sequence of four, 20, 100, 500, 2,500.
Or we could have had a common ratio of negative five, giving us the sequence four, negative 20, positive 100, negative 500, positive 2,500.
Sadly, we've reached the end of the lesson, but in summary, we can recognise geometric sequences by finding a common ratio between successive terms. For example, seven, 14, 28, 56 is geometric because it has a common ratio of two between successive terms. Hope you've enjoyed this lesson as much as I have, and I look forward to seeing you again soon for more mathematics, goodbye for now.
