Lesson video
In progress...Loading...
Hello, Mr. Robson here again.
Lovely of you to join me today.
Sequences, finding the nth term of arithmetic ones.
Oh yes, let's get learning.
For learning outcome is that we'll be able to find the nth term by investigating the common difference.
Just to remind you what the nth term is, because I'm using that phrase a lot this lesson.
Let's start by finding the nth term of increasing sequences.
Here are the first 5 terms are some arithmetic sequences.
2n, 3n, 4n.
How are they different to each other? Exactly.
There's 2n sequences, a common difference of 2.
3n a common difference of 3.
4n a common difference of 4.
2n goes up in 2's.
3n goes up in 3's.
4n goes up in 4's.
So, what would 5n look like? Absolutely.
It would go 5, 10, 15, 20, 25.
I hope you were saying that along with me.
Why does it do that? Find the first term, find the second term, find the third term, find the fourth term, find the fifth term.
How many 5's have we got? It's your 5 times table, the 5n sequence.
With what you've seen there could you fill in the blanks for these arithmetic sequences? Pause and give this a go.
Absolutely.
You should have spotted that first one is 6, 12, 18.
24, 30.
It's an arithmetic sequence between 6 and 12, a difference of 6.
We go up in 6's, we get that, our 6 times table, the sequence 6n.
The second one.
Well, if 6n goes up in 6's, 7n goes up in 7's.
And then the next one's going up in 8's.
That's gonna continue like that and be the sequence 8n.
10 to 30 in 2 steps.
That's steps of 10.
10, 20, 30, 40, 50.
That's gonna be 10n.
Absolutely.
Bottom one's a little bit trickier.
51 to 68.
Gap of 17.
Just continue on.
Another 17 on.
17 before.
17 before that.
That's 17n.
So how do we apply this to trickier sequences? We go back to something really easy that I know you know.
What happens when you add 2 to 5? You get 7.
I know that you know that, but have you thought of it like this? When you add 2 to 5 it's a translation.
You move from the position 5 2 in the positive direction to 7.
Think of that as a translation.
That's gonna help us find the nth term of trickier sequences.
What's the same, what's different about the sequences 5n and 5n plus 2? That's the sequence 5n.
That's the sequence 5n plus 2.
I'd like you to tell me something that's the same, something that's different.
Pause this video and say some suggestions to the screen.
I hope you said they've both got a common difference of 5.
They're 5n sequences.
They've both got a common difference of 5, but one starts at 5 and one starts at 7 before they start to go up in their 5's.
Hmm.
Term by term, what is the difference between 5n and 5n plus 2? Term by term, if I line them up, line up those first terms, line up those second terms, the third terms, the fourth, what is the difference? I hope you are shouting at the screen now, "2." There's a difference of 2 there.
There's a difference of 2 there.
There's a difference of 2 there.
There's a difference.
Between the respective terms I have to add 2 to get from the sequence.
5n to the sequence 5n plus 2.
Are you surprised by that? To get from 5n to 5n plus 2 I just have to add 2 to all the terms? Why is that difference of 2 there? If I said find the first 5 terms of 5n, you'd do something like this, you'd substitute in n equals 1, n equals 2, n equals 3, n equals 4, n equals 5.
You generate the terms that way.
5 lots of 1, 5 lots of 2, 5 lots of 3.
5n plus 2 you do the same.
It's just the arithmetic is a little bit different.
5 lots of 1 plus 2 makes 7, 5 lots of 2 plus 2 makes 12, et cetera.
Can you see where the extra 2 came from? It's just a 5n sequence with another 2 each time.
Another way of thinking of this, back to that notion of translation, the sequence 5n lives there on a number line.
The sequence 5n plus 2 lives there on a number line, but jumping up the same step each time.
The 5n sequences a difference of 5, except 5n plus 2 is a translation by 2 in the positive direction.
Your turn.
Pause this video.
We should have got those terms for 3n, and you should have drawn it there.
3n plus 1 drawn there.
3n plus 2 drawn there.
3n plus 3 drawn there.
What do they all have in common? Common difference of 3.
Common difference of 3.
Common difference of 3.
Common difference of 3.
I should probably say common difference of positive 3.
Oh, 3n sequences have a common difference of 3.
Who'd have thought it? What about the translations? From 3n, how did we get to the sequence 3n plus 1? Well, 3n is not a translation of 3n, it just is 3n.
So we could say it's no translation from 3n, or we could say it's 3n plus nothing, but 3n plus 1 is a translation of the sequence 3n by positive 1.
3n plus 2 is a translation by positive 2.
3n plus 3 is a translation by positive 3.
3n plus 4 would be a translation by? That's right, positive 4.
So this is how we find the nth term of any given arithmetic sequence.
5, 9, 13, 17, 21.
I need 2 things.
I need to know the common difference and I need to know the translation.
So if I map that out in a number line, I can see the common difference.
Yep, 4.
That tells me that this is a 4n sequence, but it is not the sequence 4n.
The sequence 4n goes 4, 8, 12, 16.
Our sequence isn't in that position.
It's a translation of the 4n sequence.
By how much? You can see it, can't you? It's a translation by positive 1.
So we call this sequence 4n plus 1.
It's our 4n sequence translated 1 in the positive direction.
Without a number line we compare terms in the respective positions.
So I need the common difference, I need the translation.
I know there's a common difference of 4, makes it a 4n sequence.
If I write the sequence 4n above it I can then line up those first terms, 4 and 5, line up the second terms 8 and 9, line up the third terms 12 and 13.
What's the difference? What's the translation? Well, in each case here we've got a translation of positive 1.
To get from the sequence 4n to this sequence we need to add 1.
Hence we call this sequence 4n plus 1.
Right, I'm gonna practise one.
Show you how it's done, again, and then I'm gonna ask you to try one.
Find the nth term.
8, 14, 20, 26, 32.
Okay, I need the common difference and the translation.
Have I said that enough yet? Common difference and translation.
That's all we need.
Common difference.
I'm going up in steps of 6.
That's gonna make this a 6n sequence.
I'm gonna line my sequence 6n alongside the sequence we're trying to find the nth term for.
I need to compare terms with the same term number, compare my first term, 6, to the first term, 8.
Second term, 12, to the second term, 14.
That shows me this translation.
It's positive 2 to get from 6 to 8.
I'm just gonna check it works for every term.
It does.
I'm translating by positive 2 each time.
Got a common difference of 6, a translation of 2.
That is the nth term 6n plus 2.
The coefficient of the N tells me the common difference.
The translation is that constant, positive 2.
Your turn.
Find the nth term for that one.
Pause this video, see if you can repeat that skill.
How'd we do? Two things you want? What are they? That's right, common difference and the translation.
Common difference of your sequence.
5, or positive 5 to be precise.
Common difference of positive 5 tells you it's a 5n sequence, but it isn't exactly 5n is it? It's not going 5, 10, 15, 20, 25.
When we compare the first term to the the first term, the second term to the second term, you see that you are always 3 above, or that your sequence is a translation positive 3 from the sequence 5n.
So you should have said that is the sequence 5n plus 3.
If you did, give yourself a pat on the back.
Next up, Aisha and Sofia are discussing the arithmetic sequence with the nth term 10n minus 3.
"I see a negative 3, so I think this sequence decreases." Interesting.
Sofia says, "I think it starts at negative 3 and goes up by 10 each time." Hmm.
Do you agree with either student? Pause this video, tell the person next to you.
We can check by generating the first few terms. So n equals 1, 10 lots of 1 minus 3, that's 7.
10 lots 2 minus 3, 17, et cetera, et cetera.
So we know that this sequence, 10n minus 3, goes 7, 17, 27, 37, 47.
There it is.
So Aisha said, "I see a negative 3, so I think the sequence decreases." That's not true.
It's still increasing.
It's still got a positive 10 increase.
Funnily enough, the coefficient of n, 10, a 10n sequence, it's got a difference of positive 10 between the terms, so it's not decreasing, it's still got a common difference of positive 10.
Sofia said, "I think it starts at negative 3 goes up by 10 each time." She's absolutely right.
It goes up at 10 each time.
However, it started at 7 because it's a translation minus 3 from 10.
What do we mean by that? Common difference is that 10 translation is a negative 3.
Every term is 3 less than the corresponding term in the 10n sequence.
10n would go 10, 20, 30, 40, 50.
Our sequence is simply 3 lower.
7, 17, 27, 37, 47.
Just wanna check you've got that.
Corresponding terms in the sequence 8n minus 5 will be a translation by how much from terms in the sequence 8n? Would it be 5, negative 5, or 8? Pause this video, tell the person next to you.
Super.
Negative 5 was the answer.
Sequence 8n begins 8, 16, 24, 32.
That's a translation of negative 5.
Another one for you.
What will the common difference be for the sequence with the nth term 5n minus 8? Is it 5, 8, or negative 8? It was A, 5.
The sequence 5n minus 3 goes negative 3, 2, 7, 12.
That's a common difference of 5.
It is the coefficient of n that tells us the common difference.
A 5n sequence will have a common difference of positive 5.
Hmm.
This is unusual.
Or is it exactly the same? Which statements are true of the sequence an plus b? Four statements there, some are true, some are not.
Can you identify which is which? Pause this video, have a little think.
The truth was in A and D.
The common difference of a, and a translation of an by b.
The coefficient of n tells us the common difference.
I do feel like I'm saying that a lot.
I hope it's sticking.
The coefficient of n tells us the common difference.
In this case it's a.
n is just the term number.
The constant is the translation.
In this case an plus b the constant is the b.
Straightforward practise.
There are some sequences.
I'd like to know the nth term of each one.
Pause this video, have a go at those.
Andeep says, "The sequence negative 5, negative 3, negative 1, 1, 3 is 2n minus 5 because the common difference is 2 and it starts at negative 5." Logical, but wrong.
I'd like you to give 2 explanations as to why Andeep is wrong.
There are multiple ways you could explain this.
I'd like you to give me two of those explanations, and I'm offering you that number line at the bottom because that might help with one of your explanations.
Pause this video and get writing.
nth term of these sequences.
Common difference, translation.
Common difference of 7, translation of positive 3 from 7n.
That is 7n plus 3.
Common difference of positive 7.
Translation of negative 6 from the sequence 7n.
That is 7n minus 6.
Common difference of positive 3.
Despite all those negative terms, it was a positive common difference.
A translation from the 3n sequence of negative 13, 3n minus 13.
Next one's tricky.
Or is it? Not really.
Common difference, positive 0.
4.
Translation from 0.
4n, positive 0.
5.
That's 0.
4n plus 0.
5.
If we can do it for decimals, we can do it for fractions.
Common difference of 1/2 to get from a 1/4, to 3/4, to 5/4, add 1/2 each time.
What's it a translation from the sequence 1/2n? Well, we're 1/4 below, so it is the sequence 1/2n minus 1/4.
Lovely.
Last up, Andeep made this statement.
We could have explained this in lots of ways.
The sequence 2n starts 2, 4, 6, but our sequence is negative 5, negative 3, negative 1.
That's a translation of negative 7 each time.
Andeep's got it down as 2n minus 5.
We know it's a translation of negative 7, so it's 2n minus 7.
You could use that explanation.
You might have shown that same translation on a number line.
The sequence 2n is there.
If we wanna get to where our sequence is, as in minus 5 going up in steps of 2, we need a translation of negative 7 so we can show visually that to get from 2n to our sequence, it's 2n minus 7, not 2n minus 5.
Finally, you might have gone with, well the first term in 2n minus 5 would be negative 3.
So that can't describe this sequence.
If the first term of 2n minus 5 is negative 3 and the first term of our sequence is negative 5, our sequence cannot be 2n minus 5.
Finding the nth term of decreasing sequences now.
Let's compare these two sequences, 10 plus 2n and 10 minus 2n.
What do you notice? Pause the video.
I hope you noticed that the sequence 10 plus 2n is increasing by 2 each time, whereas sequence 10 minus 2n is decreasing by 2 each time.
Why does one increase and the other decrease? Well, if we look at how we generate the terms of these sequences, 10 plus 2n is 10 lots of 2 plus 1, 10 lots of 2 plus 2, 10 lots of 2 plus 3.
As the sequence grows, we add more and more lots of positive 2.
By contrast 10 minus 2n, well we start by taking away one lot of 2.
And then we take away 2 lots of 2, and then we take away 3 lots of 2.
Or we could think of it as adding more lots of negative 2.
As the sequence grows, we add more and more lots of negative 2, hence it's a decreasing sequence.
So a positive coefficient of n gives us an increasing sequence.
A negative coefficient of n gives us a decreasing sequence.
If I changed 10 minus 2n to 10 minus 3n, I'd be adding more and more negative 3's making it more and more negative.
The coefficient of n determines the common difference.
If it's negative, that common difference is going to be negative.
The sequence is going to decrease.
So which of these arithmetic sequences is decreasing? Pause this video.
It was the last two, negative 8n plus 3, and 8 minus 3n.
The first one, 8n minus 3 has a positive 8 coefficient of n.
That's an increasing sequence.
The second one, positive 3 coefficient of n.
That's an increasing sequence.
Negative 8n plus 3, that's a negative 8 coefficient of n.
That's a decreasing sequence.
8 minus 3n.
We can think of that as 8 and negative 3n.
That's gonna give us a negative 3 coefficient of n decreasing that sequence.
Just to let you know, 8 minus 3n, you might see it written as negative 3n plus 8.
That's just a commutative of edition.
You might see it written that way round.
And negative 8n plus 3 you might see written as 3 minus 8n.
What's the nth term of this sequence? 100, 90, 80, 70, 60.
Sam says, "I think it starts, I think it it's a 100 minus 10n because it starts at 100 and decreases by 10 each time." That's pretty logical, isn't it? How do you know immediately that Sam is wrong? Pause this video, tell the person next to you.
We can check the first term and we know that a 100 minus 10n can't be right because when n is 1, 100 minus 10 lots of 1, that's 90.
What did Sam get right? The common difference of negative 10.
So the coefficient n will be negative 10.
But negative 10n as a sequence would be there.
It would go negative 10, negative 20, negative 30.
Where is our sequence by comparison? 100, 90, 80.
I'm almost asking you what's the translation from negative 10n to our sequence? The translation that maps the terms of negative 10n onto our given sequence is positive 110.
Minus 10 plus 110 takes us to 100.
Minus 20 plus 110 takes us to 90.
So we've got a common difference of negative 10 and a translation of positive 110.
That'll give us an nth term of 110 minus 10n, or negative 10n plus 110.
You can see the translation when you align the sequences.
If I line up the first terms, line up the second terms, line up the third terms, and then comparing negative 10 to 100, negative 20 to 90, negative 30 to 80.
And in each and every case I have a translation of 110.
So we know this is a negative 10n sequence translated by positive 110.
The common difference, the coefficient of n and our translation 110.
The coefficient of n is the common difference of the sequence, the constant is the translation.
Nothing's changed with decreasing sequences.
Still the same thing.
We're looking for the common difference and the translation.
Laura and Jacob are discussing the nth term of the sequence 15, 10, 5, 0, negative 5.
Laura says, "I think it's 20 minus 5n." Jacob says, "I think it's 15 minus 5n." Use the number line to show them who is right.
Your explanation might have included to translate the sequence negative 5n to the position of 15, 10, 5, you need a translation of positive 20.
Therefore, Laura is right.
There's the sequence negative 5n and that's the translation required to get it into the position of 15, 10, 5, 0, et cetera.
You could also check that Laura's right by testing the first term.
When n equals 1, 20 minus 5 lots of 1 makes 15.
Okay, I'm gonna do an example now and then I'll ask you to have a go at an example.
Find the nth term of negative 0.
08, negative 0.
11, negative 0.
14, et cetera.
I'm pretty sure I'm just gonna find the common difference in the translation again.
The common difference is negative 0.
03.
So my negative 0.
03n sequence would go like that.
And then what's the translation? Term by term, what is the translation? I'm lower by 0.
05.
or I have a translation of negative 0.
05.
So I've translated the sequence negative 0.
03n by negative 0.
05.
What's my nth term gonna be? Negative 0.
03n minus 0.
05.
Common difference, coefficient of the n.
Translation, the constant.
I'd like you to find the nth tern of this sequence.
Pause this video, give it a go.
So common difference in translation, nothing's changed.
That's what we're looking for.
Common difference of negative 0.
15.
Negative 0.
15n would look like that.
What's the translation? Negative 1.
95 each time.
Giving you an nth term negative 0.
15n minus 1.
95.
Practise time now.
First task, I'd like you to match the sequences to their respective nth terms. Five sequences there, 5 nth terms. Draw some lines, match them up.
For question 2 I'd like you to find the nth term of these sequences.
Thirdly, 2 students are discussing arithmetic sequences.
Aisha says, "If you give me any 2 terms of arithmetic sequence, I can tell you the nth term." Sofia responds with, "You're right, Aisha.
If we had 30 and 25, we'd know it's 30, 25, 20, 15, 10 with an nth term of 35 minus 5n." 35 minus 5n, that is the right nth term for that sequence 30, 25, 20 15.
Write a sentence to explain if Aisha and Sofia are right.
Pause this video and write a sentence.
We'll match 29 minus 5n with a sequence starting 24.
29 minus 5n lots of one starts at 24 negative 5n.
That's gonna have a common difference of negative 5.
24, 19, 14.
Common difference of negative 5.
24 minus 3n, that's gonna start at 21.
It's gonna decrease by 3 each time, and that one does.
29 minus 3n.
Negative 3n minus 2, and negative 5n plus 2 matches to negative 3, negative 8, negative 13.
The nth term of these sequences.
We've got a common difference of negative 12 and a translation of positive 30 that gives us a nth term of 30 minus 12n.
You might have written negative 12n plus 30.
Either would be acceptable.
In the second case we have a constant difference of negative 12 again, however, we've got a translation of positive 3 this time.
So you have a sequence of negative 12n plus 3.
You might have written 3 minus 12n and that will be absolutely fine.
The next sequence.
For the next sequence, 15 minus 3n.
For the decimals in E, we've got a common difference of negative 0.
3, but a translation of positive 1.
5.
So nth term of 1.
5 minus 0.
3n.
You may have written negative 0.
3n plus 1.
5.
Either is acceptable.
And then sequence F, I see that decreasing by 1/2 each time.
Or a common difference of negative 1/2.
And a translation of positive 4.
So it's positive 4 minus 1/2n, or negative 1/2n plus 4.
Okay, in the case of Aisha and Sofia, initially they look correct.
The sequence 30, 25, 20.
That will have a nth term of 35 minus 5n.
However, we don't which 2 terms they are.
30 and 25 could be anywhere in that sequence.
They might be the 15th term and the 16th term.
So we need to know that they're the first and the second terms in order to know that that is the nth term 35 minus 5n.
Finding and using the nth term now.
An arithmetic sequence starts negative 7, negative 5, negative 3, negative 1, 1.
What term number is 399? Interesting.
Well, we know that this is a sequence increasing by 2 each time.
We know there's a translation of negative 9 from the 2n sequence.
We know the nth term is 2n minus 9.
2 times something minus 9 gives us 399.
In order to find the term number 399, 2 lots of something minus 9.
We can start with approximation.
How do we get near that term value of 399? Let's start with 2 lots of 200 minus 9 'cause that gets us incredibly close.
If we take the 200th term, 2 lots of 200 minus 9, we have a term of 391.
We can count on from there.
If 391's a 200th term in the sequence 2n minus 9, we know it's arithmetic, it's gonna go up by 2 every time, and we count until we hit 399.
Quick question for you.
We want to know which term 243 is in the arithmetic sequence 5n minus 7.
Which of the below is a sensible approximation? Is it 243, n equals 50, or n equals 100? Pause this video, tell the person next to you.
It is n equals 50.
Lots of 50 gets us to 250, which is really close to our term 243.
5 lots of 100, too much.
5 lots of 243 way too much.
And arithmetic sequence starts with the terms. 89.
625, 89.
275, 88.
925.
When will it reach 21.
025? For this one, we're gonna want our calculators.
Our calculator's going to help us to find the common difference and the translation.
The common difference, just take any term and take away the previous term, you'll find the common difference.
Negative 0.
35.
To find the translation, our negative 0.
35n sequence starts with negative 0.
35.
The difference between our first term, 89.
65 and negative 0.
35 will be 89.
975.
We've got translation of 89.
975, giving us an nth term of 89.
975 minus 0.
35n.
A test just to check that that is our end term when n equals 1.
Yes, 89.
975 minus 1 lot at 0.
35 gives us our first term.
So that works.
Let's just check the fifth term.
It worked again.
Really importantly, did you know that you can use the left arrow keys to change the term number? I pressed Left, Left, Delete, and then put a 5 in place of the 1 to change from my first term number to my fifth term number.
If you didn't know how to do that, just pause this and have a little practise for your calculator and see if you can make it do the same thing using your Left Arrow key and the Delete button.
So when will it reach 21.
025? Let's use trial and improvement.
What about when n equals 100? Again, I've Left Arrow key'd and substituted in my 100 into the n position.
I got to 54.
975.
Not to close enough.
I'm gonna use my left arrow key to make my calculator look like that and then I'm gonna substitute in n equals 200.
See if we get any closer.
19.
975.
I am a lot closer.
What about 199? 20.
325.
198.
20.
675.
197, 21.
025.
When will that sequence reach 21.
825? At the 197th term.
Quick check for you.
Which of the below finds the term negative 60.
67 in the sequence 8.
7 minus 0.
07n? That's right, it's the third one.
Question one, find the nth term.
Use approximation and counting on to find the term 705 in this arithmetic sequence.
Question 2, very similar.
Find the nth term.
Use approximation and counting on to find the term negative 554.
And question 3, Jacob wants to know the last positive term in the sequence 36.
95 minus 0.
06n.
Use trial and improvement on your calculator to help him.
Pause this video.
Okay, finding the nth term using approximation and counting on, the nth term is 10n minus 35.
We can approximately find from the 70th term, which is 665, we can count on in 10's, it's the 74th term in the sequence.
Really similar question, two.
Just a slightly different nth term.
10 minus 12n is the nth term.
Negative 554, 47th term in the sequence.
Finally the last positive term in 36.
95 minus 0.
06n.
So approximation.
If I go 500 but it's not enough, I need to take more away.
So left arrow keys, change that up to 600.
Still not quite there.
Let's try 610.
Nearly there.
620.
Oof, we're into the negatives now.
So somewhere between the 610th and the 620th, the 615th, the 616th term.
That's interesting.
The 615th term is positive.
The 616th term is negative.
Therefore the 615th term is the last positive.
So, to summarise.
We find the nth term of arithmetic sequences by finding the common difference between the terms. For example, 5, 8, 11, 14, 17, a common difference of 3.
Therefore, it's a translation of the sequence with the nth term 3n.
So translation by positive 2 of the sequence, therefore the nth term is 3n plus 2.
I hope you've enjoyed today's lesson learning about nth terms and sequences.
I look forward to seeing you again soon for more mathematics.
