Lesson video
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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.
So 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 five terms of some arithmetic sequences: 2n, 3n, 4n.
How are they different to each other? Exactly, the 2n sequence has a common difference of two, 3n a common difference of three, 4n a common difference of four.
2n goes up in twos, 3n goes up in threes, 4n goes up in fours.
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 fives have we got? It's your five 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 six and 12, a difference of six.
We go up in sixes.
We get that, our six times table, the sequence six and the second one.
Well, six angles up in sixes, seven angles up in sevens, and then the next one's going up in eights.
That's gonna continue like that and be the sequence.
8n, 10 to 30 and two 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, 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 two to five? You get seven.
I know that you know that, but have you thought of it like this? When you add two to five, it's a translation.
You move from the position five, two in the positive direction to seven.
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 two? That's the sequence 5n.
That's the sequence 5n plus two.
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 five.
They're 5n sequences.
They've both got a common difference of five, but one starts at five and one starts at seven before they start to go up in their fives.
Term by term, what is the difference between 5n and 5n plus two? 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 two.
There's a difference of two there.
Between the respective terms, I have to add two to get from the sequence 5n to the sequence 5n plus two.
Are you surprised by that? To get from 5n to 5n plus two, I just have to add two to all the terms. Why is that difference of two there? If I said find the first five terms of 5n, you'd do something like this.
You'd substitute in n equals one, n equals two, n equals three, n equals four, n equals five.
You generate the terms that way.
Five lots of one, five lots of two, five lots of three.
5n plus two, you do the same.
It's just your arithmetic looks a little bit different.
Five lots of one plus two makes seven.
Five lots of two plus two makes 12, et cetera.
Can you see where the extra two came from? It's just a 5n sequence with another two 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 two lives there on a number line.
But jumping up the same step each time, the 5n sequences, a difference of five except 5n plus two is a translation by two 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 one drawn there.
3n plus two drawn there.
3n plus three drawn there.
What do they all have in common? Common difference of three.
I should probably say common difference of positive three.
Oh, 3n sequences have a common difference of three.
Who'd have thought it? What about the translations? From 3n, how did we get to the sequence 3n plus one? 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 one is a translation of the sequence 3n by positive one.
3n plus two is a translation by positive two.
3n plus three is a translation by positive three.
3n plus four would be a translation by? That's right, positive four.
So this is how we find the nth term of any given arithmetic sequence.
5, 9, 13, 17, 21.
I need two 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, four, 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 one.
So we call this sequence 4n plus one.
So we call this sequence 4n plus one.
It's a 4n sequence translated one in the positive direction.
Without a number line, we compare terms in their respective positions.
So I need the common difference.
I need the translation.
I know there's a common difference of four.
Makes it a 4n sequence.
If I write the sequence 4n and above it, I can then line up those first terms four and five, line up the second terms eight and nine, 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 one.
To get from the sequence 4n to this sequence, we need to add one.
Hence we call this sequence 4n plus one.
All 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 in translation.
That's all we need, common difference.
I'm going up in steps of six.
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 six to the first term eight, second term 12 to the second term 14.
That shows me this translation.
It's positive two to get from six to eight.
I'm just gonna check it works for every term.
It does, I'm translating by positive two each time.
Got a common difference of six, a translation of two.
That is the nth term 6n plus two.
The coefficient of the n tells me the common difference.
The translation is that constant positive two.
Your turn, find the nth term for that one.
Pause this video.
See if you can repeat that skill.
How do we do? Two things you want, what are they? That's right, common difference and the translation.
Common difference of your sequence.
Five or positive five to be precise.
Common difference of positive five tells you it's a 5n sequence, but it isn't exactly 5n, is it? It's not getting 5, 10, 15, 20, 25.
When we compare the first terms, the first term, the second term, second term, you see that you're always three above or that your sequence is a translation positive three from the sequence 5n.
So you should have said that is the sequence 5n plus three.
If you did, give yourself a pat on the back.
Next up, Aisha and Sophia are discussing now with arithmetic sequence with the nth term 10n minus three.
"I see a -3, so I think "this sequence decreases, interesting." Sophia says, "I think it starts at -3 "and goes up by 10 each time." 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 one, 10 lots of one minus three, that's seven.
10 lots of two minus three, 17, et cetera, et cetera.
So we know that this sequence 10n minus three goes 7, 17, 27, 37, 47.
There it is.
So I should said, I see a -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.
Sophia said, "I think it starts at -3 "and goes up by 10 each time." She's absolutely right.
It goes up at 10 each time.
However, it started at seven because it's a translation minus three from 10.
What do we mean by that? Common difference is that 10, translation is a -3.
Every term is three less than the corresponding term in the 10n sequence.
10n would go 10, 20, 30, 40, 50.
Our sequence is simply three lower, 7, 17, 27, 37, 47.
Just wanna check you've got that.
Corresponding terms in the sequence 8n minus five will be a translation by how much from terms in the sequence 8n? Would it be five, negative five, or eight? Pause this video, tell the person next to you.
Super, negative five was the answer.
Sequence 8n begins 8, 16, 24, 32.
Sequence 8n minus five begins 3, 11, 19, 24.
That's a translation of -5.
Another one for you.
What will the common difference be for the sequence with the nth term 5n minus eight? Is it five, eight, or negative eight? It was a.
five.
The sequence 5n minus three goes -3, 2, 7, 12.
That's a common difference of five.
It is the coefficient of n that tells us the common difference.
A 5n sequence will have a common difference of positive five.
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, a 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.
Practise time now.
Match the nth term with the sequences in the table.
Seven sequences in rows in that table.
The top row goes 4, 8, 12, 16, 20.
That sequence can be described by one of the term expressions on the left-hand side.
Mr. Robson, you've given us seven sequences, seven rows, but nine expressions.
That's right, some of those expressions will not fit in the table.
I'd like you to identify which ones don't fit in the table as well.
Pause this video, give this problem a go.
Question two, 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.
Question three, Andeep says, "The sequence -5, -3, -1, 1, 3 "is 2n minus five "because the common difference is two "and it starts at -5." Logical but wrong.
I'd like you to give two 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.
Feedback time now.
Match the nth terms with the sequences in the table.
4n, 8n, we saw those right at the start of the lesson.
I hope they jumped out to you immediately.
That's a translation positive three from the sequence 4n.
A translation -1 from the sequence 8n.
The next one was 4n plus four, 8n minus seven, and 4n minus 14.
So which ones did not match? 4n minus 10 and 8n plus one.
How did we know? Well, you can find the first term.
So we knew that those two did not match with what we had in the table.
Question two, at the term of these sequences, common difference, translation, common difference of seven, translation of positive three from 7n.
That is 7n plus three.
Common difference of positive seven.
Translation of -6 from the sequence 7n.
That is 7n minus six.
Common difference of positive three.
Despite all those negative terms, it was a positive common difference.
A translation from the 3n sequence of -13, 3n minus 13.
Next one's tricky, or is it? Not really, common difference positive 0.
4.
Translation from 0.
4n, that's 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 a half to get from a quarter to three quarters to five quarters, add a half each time.
What's the translation from the sequence half n? Well, we're a quarter below, so it is the sequence half n minus a quarter, 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 -5, -3, -1.
That's a translation of -7 each time.
Andeep has got it down as 2n minus five.
We know it's a translation -7, so it's 2n minus seven.
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 five going up in steps of two, we need the translation of -7 so we can show visually that to get from 2n to our sequence.
It's 2n minus seven, not 2n minus five.
Finally, you might have gone with, well, the first term in 2n minus five would be -3.
So that can't describe this sequence.
If the first term of 2n minus five is -3 and the first term of our sequence is -5, our sequence cannot be 2n minus five.
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 two each time, whereas sequence 10 minus 2n is decreasing by two 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 two plus one, 10 lots of two plus two, 10 lots of two plus three.
As a sequence grows, we have more and more lots of positive two.
By contrast, 10 minus 2n, well, we start by taking away one lot of two, and then we take away two lots of two, and then we take away three lots of two, or we could think of it as adding more lots of -2.
As a sequence grows, we add more and more lots of -2, hence it's 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 -3s 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, <v ->8n plus three and eight minus 3n.
</v> The first one, 8n minus three has a positive eight coefficient of n.
That's an increasing sequence.
The second one, positive three coefficient of n.
That's an increasing sequence.
<v ->8n plus three, that's a -8 coefficient of n.
</v> That's a decreasing sequence.
Eight minus 3n, we can think of that as eight and -3n.
That's gonna give us a -3 coefficient of n, decreasing that sequence.
Just to let you know, eight minus 3n, you might see it written as -3n plus eight.
That's just a commutative of addition.
You might see it written that way round.
And -8n plus three, you might see written as three minus 8n.
What's the nth term of this sequence? 100, 90, 80, 70, 60.
Sam says, "I think it's 100 minus 10n "because it starts at a 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 100 minus 10 lots of one, that's 90.
What did Sam get right? Common difference of -10.
So the coefficient of n will be -10.
But -10n as a sequence would be there.
It would go -10, -20, -30.
Where is our sequence by comparison? 100, 90, 80.
I'm almost asking you what's the translation from -10n to our sequence? The translation that maps the terms of -10n onto our given sequence is positive 110 minus 10 plus 110 takes us to a 100 minus 20 plus 110 takes us to 90.
So we've got a common difference of -10 and a translation of positive 110.
That'll give us an nth term of a 110 minus 10n or -10n plus a 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 -10 to a 100, <v ->20 to 90, -30 to 80,</v> and in each and every case, I have a translation of a 110.
So we know this is a -10n sequence translated by positive 110.
The common difference, the coefficient of n, and our translation a 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, -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 -5n to the position of 15, 10, 5.
You need a translation of positive 20.
Therefore, Laura is right.
There's the sequence -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 is right by testing the first term.
When n equals one, 20 minus five lots of one makes 15.
Okay, I'm gonna do an example now and I'll ask you to have a go at an example.
Find the nth term of -0.
08, <v ->0.
11, -0.
14, et cetera.
</v> I'm pretty sure I'm just gonna find the common difference in the translation again.
The common difference is -0.
03.
So my -0.
03n sequence would go like that.
And then what's the translation term by term? What is the translation? I'm lower by five hundredths or I have a translation of -0.
05.
So I've translated the sequence <v ->0.
03n by -0.
05.
</v> What's my nth term gonna be? <v ->0.
03n mminus -0.
05.
</v> Common difference, coefficient of n.
Translation, the constant.
I'd like you to find the nth term 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 -0.
15.
<v ->0.
15n would look like that.
</v> What's the translation? <v ->1.
95 each time</v> giving you an nth term -0.
15n minus 1.
95.
giving you an nth term -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, five nth terms. Draw some lines, match them up.
For question two, I'd like you to find the nth term of these sequences.
Thirdly, two students are discussing arithmetic sequences.
Aisha says, "If you give me any two terms "of arithmetic sequence, I can tell you the nth term." Sophia 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 Sophia are right.
Pause this video and write a sentence.
We'll match 29 minus 5n with a sequence starting 24.
29 minus five lots of one starts at 24.
<v ->5n, that's gonna have a common</v> difference of -5, 24, 19, 14.
Common difference of -5.
24 minus 3n, that's gonna start at 21.
It's gonna decrease by three each time and that one does.
29 minus 3n, -3n minus two, and -5n plus two matches to -3, -8, -13.
The nth term of these sequences, we've got a common difference of -12 and a translation of +30, that gives us the nth term of 30 minus 12n.
You might have written -12n plus 30, either would be acceptable.
In the second case, we have a constant difference of -12 again.
However, we've got a translation of positive three this time.
So you have a sequence of -12n plus three.
You might have written three minus 12n and that will be absolutely fine.
The next sequence, -12n, -3.
For the next sequence, 15 minus 3n.
For the decimals in E, we've got a common difference of -0.
3, but a translation of +1.
5.
So an nth term of 1.
5 minus 0.
3n.
You may have written -0.
3n plus 1.
5, either is acceptable.
And then sequence F, I see that decreasing by a half each time, or common difference of negative a half and a translation of positive four.
So it's positive 4 minus a half n or negative a half n plus four.
Okay, in the case of Aisha and Sophia, initially, they look correct.
The sequence 30, 25, 20, that will have an nth term of 35 minus 5n.
However, we don't know which two 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 -7, -5, -3, -1, 1.
What term number is 399? Interesting, well, we know that this is a sequence increasing by two each time.
We know there's a translation of -9 from the 2n sequence.
We know the nth term is 2n minus nine.
Two times something minus nine gives us 399.
In order to find the term number 399, two lots of something minus nine.
We can start with approximation.
How do we get near that term value of 399? Let's start with two lots of 200 minus nine 'cause that gets us incredibly close.
If we take the 200th term, two lots of 200 minus nine, we have a term of 391.
We can count on from there.
If 391 is a 200th term in the sequence 2n minus nine, we know it's arithmetic.
It's gonna go up by two 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 seven.
Which of the below is a sensible approximation? Is it 243, n equals 50, or n equals a 100? Pause this video, tell the person next to you.
It's n equals 50.
Five lots of 50 gets us at 250, which is really close to our term 243.
Five lots of a 100, too much.
Five lots of 243, way too much.
An 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 is going to help us to find the common difference in the translation.
The common difference, just take any term and take away the previous term.
You'll find the common difference -0.
35.
To find the translation, our -0.
35n sequence starts with -0.
35.
The difference between our first term 89.
625 and -0.
35 will be 89.
975.
So we've got a translation of 89.
975 giving us the nth term of 89.
975 minus -0.
35n.
A quick test just to check that that is our nth term.
N equals one, yes.
89.
975 minus one lot of -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 five in place of the one 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 calculation.
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 a 100? Again, I've left arrow key and substituted in my 100 into the nth position.
I got to 54.
975, not 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.
198, 20.
675, 197, 21.
025.
When will that sequence reach 21.
025 at the 197th term? Quick check view, which of the below finds the term <v ->60.
67 in the sequence 8.
7 minus -0.
07n?</v> <v ->60.
67 in the sequence 8.
7 minus -0.
07n?</v> 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 two, very similar.
Find the nth term, use approximation, counting on to find the term -554.
And question three, 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, find 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 10s.
It's the 74th term in the sequence.
Really similar question to just a slightly different nth term.
10 minus 12n is the nth term.
<v ->554, 47th term in the sequence.
</v> 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, ooh, we're into the negatives now.
So somewhere between the 610th and the 620th, the 615th, the 616th term, that's interesting.
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 one.
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 three.
Therefore, it's translation of the sequence with the nth term 3n.
So translation by positive two of the sequence.
Therefore, the nth term is 3n plus two.
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.
