Lesson video

Hello.

Mr. Robson here.

Lovely to see you again.

Great choice to join me for maths sequences and this topic within sequences is just delightful.

I hope you enjoy it as much as I do.

A learning outcome for today is we'll be able to determine whether a number is a term in a given arithmetic sequence.

So it would be sensible to look again at the definition of an arithmetic sequence because there are keywords you're gonna hear a lot today.

An arithmetic, also known as linear sequence, is a sequence where the difference between successive terms is a constant.

An example 5, 9, 13, 17 is an arithmetic sequence, a constant difference of positive 4 between the terms, whereas 1, 2, 4, 8 is not arithmetic.

The difference between the terms is not constant.

Throughout the lesson today, we're gonna justify terms of a sequence in several different ways.

The first way is that we're going to use the features of the sequence.

Alex and Andeep are asked if the number 868 is in the arithmetic sequence starting 9, 11, 13, 15, 17.

Alex says, "I'm gonna continue the sequence.

9, 11, 13, 15, 17, 19, 21, 23, 25, 27." Andeep says, "Stop.

I've noticed something about the numbers you've written, Alex.

They are different to 868." What's wrong with Alex's method and what do you think Andy has spotted? Pause this video.

Make that suggestion to the person next to you.

What's wrong with Alex's method is, it's very time inefficient.

868, we're gonna go, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41.

We could be here a while.

We haven't got all that time.

We're seeking efficiency.

We mathematicians are always seeking efficiency.

So what's Andeep spotted that could help us be more efficient? Hmm, you are right.

868 is even whereas 9 11, 13, 15, they're all odd.

So we don't need to list this sequence.

We can use the features of an arithmetic sequence to justify whether a given number is a term in that sequence or not.

Will 4,583 be a term in the arithmetic sequence starting 2, 8, 14, 20, 26? How might we justify our answer? You'll notice I've highlighted the word, justify, there a hugely important skill in mathematics.

We mathematicians, we're always seeking the truth.

Once we know the truth, we're looking to justify how do we know that is the truth? How would we explain to other people that that is the truth? That's why justifying is really important in mathematics.

Anyway, you will turn to do a little thinking.

Will 4,583 be in that sequence? And how might you justify your answer? Make some suggestions to the person next to you.

4,583 will not be in the sequence because it's an odd number.

We can't just say that.

We need a bit more detail.

We would say 2, 8, 14, 20, 26 are all even numbers.

The common difference is even, so the terms will always remain even whereas 4,083 is odd.

So it will not appear in this sequence.

Right.

Just to check you've got that, true or false.

13,913 is in the arithmetic sequence starting 32, 36, 40, 44.

Is that true or is that false? Give that answer now.

Well done.

It's false.

We can see it's not.

But how are you going to justify your answer.

Justify.

Will you say 13,913 is odd or will you say 32, 36, 14 44 are all even numbers.

The sequence is an even common difference, meaning the terms will always be even.

13,000 113 is an odd number, so it cannot be in the sequence.

Choose one of those two justifications.

Well done.

You went for the second one.

The first one, not a deep enough justification.

Look at the precision in the language on option B.

That's a much better justification.

Will 122,274 be a term in the arithmetic sequence starting 2, 7, 12, 17, 22? This is interesting because this sequence goes even, odd, even, odd.

And I definitely don't wanna keep counting up to 122,274.

Hmm.

Are there any patterns in that sequence that help you to see whether that will be a term.

Any patterns 2, 7, 12, 17? Make some suggestions to the person next to you.

So in order to justify our answer on this occasion, we need something more than odd versus even.

The sequence 2, 7, 12, 17 22 has a common difference of 5, meaning it's a translation negative 3 of the sequence 5n, our 5 times table, therefore it will always end in the digits 2 and 7.

The number 122,274 ends in a 4.

We need a 2 or a 7 in the one's column in order for it to be in the sequence.

That number does not have a 2 or a 7 in the one's column.

Therefore, it's not in that sequence.

Just to check you've got this, true or false.

897 is in the arithmetic sequence starting negative 16, negative 6, 4, 14.

True or false? Tell me now.

You should have gone with false.

Now how might we justify that 897 is not in that arithmetic sequence? Your choices are 897 ends in 7, not 4 or the sequence has a common difference of positive 10.

Therefore all subsequent positive terms will end with a 4 in the one's column.

897 does not have a one's digit of 4.

Which is the better justification? Absolutely, it's the second one.

Look at the detail.

Look at the precision in the language.

897 ends in a 7, not a 4.

That's not a detailed enough justification and it's as important to justify our answer as it is to get the answer correct.

Next, will the arithmetic sequences 5n plus 13 and 10n minus 28 ever share a common term? That feels like a huge problem.

Will they ever share a common term? Hmm.

There's lots of ways we might show this mathematically.

However, a simple way is to consider the features of the first few terms and then use those first few terms to justify the truth behind whether they share a common term or not.

So there's the first few terms of 5n plus 13, there's the few first few terms of 10n minus 28.

Look at the features of the two sequences.

You should have noticed the sequence 5n plus 13 is a translation of the sequence 5n.

Therefore we'll always end with a 3 or 8 in the one's position.

The sequence 10n minus 28, is a translation of the sequence 10n.

Therefore, all the positive terms will continue to end with the 2 in the one's position.

So it is going to be impossible for those two sequences to meet.

They will not share a common term.

Let's just check you've got that now.

Which statement is true of the sequences 5n minus 4 and 20n minus 1.

Is it that they share a common term, they do not share a common term or we can't tell if there's a common term or not? Which one of those three options do you think it is? Pause this video.

Make a suggestion to the person next to you.

We should have gone with, they do not share a common term.

If we look at the first few terms in each sequence, we get those terms 5n minus 4 is a translation by negative 4 of 5n.

Therefore, it'll always end in a 1 or 6.

Whereas 20n minus 1, translation minus 1 of the sequence 20n, it's always going to end in a 9.

Therefore, they will not share a common term.

Next.

Sometimes the justification can be about increasing or decreasing sequences.

If you look at this problem, will 16.

879 ever be in the arithmetic sequence starting 11.

758, 11.

635, 11.

512, 11.

389 et cetera? No, of course it won't.

That is decreasing sequence and 16.

879 is greater than the first term.

So it will never get to 16.

879.

Let's just check you've got that.

19/6 is in the arithmetic sequence, 17/6, 8/3, 15/6, 7/3, 13/6.

Is that true or is it false? And then can you justify your answer with either, we could make a common denominator of 6, so it might be in the sequence, or this is a decreasing sequence and 19/6 is already higher than the first term, therefore it cannot be in the sequence.

True or false? And which justification will back up your answer? I hope you went with false.

And your knowledge of your fractions should have told you that 19/6 is already higher than the first term and we're looking at a decreasing sequence, so it cannot be in that sequence.

Practise time now.

Question 1.

Will 467 be in the arithmetic sequence starting negative 6, 2, 10, 18? Will it? Will it not? That's any part of your answer.

Importantly, I'd like you to justify your answer.

That's gonna require you writing a sentence.

For question 2, 113 and 129 are consecutive terms in aromatic sequence.

Will 12,718 be the sequence? Yes or no? More importantly, write a sentence to justify your answer.

Pause this video and do that now.

Question 3, will 467 be in the arithmetic sequence starting negative 6, negative 1, 4? Again, yes or no, followed by a sentence to justify your answer.

113 and 123 are consecutive terms in arithmetic sequence.

Will 12,713 be in the sequence, yes or no? More importantly, a sentence to justify your answer.

Pause this video.

Give that a go.

Question 5 to finish.

Will the arithmetic sequences 1,234 minus 5n and 15n plus 1 ever have a common term? Again, not just yes or no.

It's a sentence or sentences to justify your answer.

Feedback now.

467 will not be in that sequence because this is an odd number.

Negative 6, 2, 10, and 18 are all even numbers and the common difference is even therefore the terms will always remain even.

For question 2, 12,718 will not be in the sequence because it's an even number.

113 and 129 are both odd and there's an even difference.

An odd plus an even will make an odd, therefore the terms will always remain odd.

They'll never land on an even number.

For question 3, 4 subtract negative 1 or 4 minus negative 1 gives you positive 5.

A common difference of 5.

See this sequence continue 4, 9, 14, 19 as a translation of the sequence of 5 and it's always going to land on a 4 or a 9 or rather it's gonna have a 4 or a 9 in the one's column.

Therefore 467 cannot be in the sequence because it's got a 7 in the one's column not a 4 or a 9.

For question 4, 113, 123 are consecutive terms of arithmetic sequence.

If that's the case, then 123 minus 113, that is 10, a common difference of 10 sees a sequence continue.

123, 133, 140, 150, 163.

A translation of the sequence 10n.

Every term is three more than a multiple of 10.

Therefore, every number 113 or greater that ends with a one's digit of 3 will be in the sequence that includes our number 12,713.

For question 5, we'd have looked at the first few terms of each sequence.

1,229, 1,224, 1,219, et cetera.

As a translation of the sequence negative 5n, it's always going to end in a 9 or a 4 until it becomes negative.

For 15n plus 1, you can see it ends in 6, it ends in 1, it ends in 6, it ends in 1.

It's a translation of the sequence, 15n, it's always going to end in a 6 or a 1 and only ever have positive terms. Therefore, those two sequences will not share a common term.

There's any moments you missed from that justification because there's a lot there.

Pause this video and make any notes that you need to make to just fill in the gaps in your justification.

Next, using a graphical representation, justifying the terms of a sequence using a graphical representation.

Our own students are discussing an arithmetic sequence.

I've worked out their first 10 terms of 91 minus 17n.

They are 74, 57, 40, 23, et cetera.

That's Alex's work.

And then, Aisha comes in.

"I've checked the 10th term." For the 10th term, n equals 10.

And when you substitute that in 91 minus 17 lots of 10, we end up with negative 79, whereas Alex has got the 10th term as negative 81.

So Aisha rightly says, "I think you've made a mistake somewhere." Aisha is correct, but can you spot where Alex has made his error? Pause this video.

Have a good look.

Can you spot Alex's error? You might have noticed it's in this moment here, we're moving in steps of negative 17 in this sequence and 6, takeaway 17 does not land us on negative 13.

It should land us on negative 11.

So negative 13 was his first error, but because he successfully subtracted 17 for all the next ones, all the subsequent terms are off.

So every term, the sixth, the seventh, the eighth, the ninth, the 10th, they're all incorrect.

Not an easy error to spot.

However, we could have made it easier to spot if we'd graphed Alex's terms, we would've seen that moment far easier.

Let's have a look at that graph.

Plotting the first term, 174, the second term, 257, 340.

Look at the graph.

We can see the moment from the sixth term on where the sequence does not align.

It's an arithmetic sequence, a linear sequence.

When we graph this sequence, the points should align.

It's that moment there where they no longer align.

All those positive terms make a perfect line and then something goes amiss at that moment.

So we could have spotted that it was that moment between the fifth and sixth term where it had gone wrong.

To check, you've got that now.

Jun has written the first 10 terms of the sequence, negative 3.

486 plus 0.

72n.

I won't read those for you.

That's what Jun's wrote down.

He then graphs them and notices he's gotten one term wrong.

Which term is it? Pause this video and make a suggestion to the person next to you.

You should have noticed the one that was not in line, Arithmetic sequence, a linear sequence, it should align.

However, on this graph, that point did not align.

Which one is that, first, second, third, fourth, fifth, sixth, seventh.

It's the seventh term which is not in line.

So we look at that seventh term, 1.

094.

That must not be the value, the true value of that seventh term 'cause when we plot 7, 1.

094, it's not in line with the others.

Practise time now.

Question one.

Sam is asked to continue this arithmetic sequence and plot it.

127,850, 135,350, 142,850.

Sam writes the below values and plots them.

Okay, we've got the values that Sam worked out and the plot.

Write a sentence explaining how you know there has been an error.

Pause this video and write that sentence.

For question 2.

Use these graphs to spot the incorrect terms in these arithmetic sequences.

I haven't given you the nth term, I've just given you the term values and the respective graph.

How do you know which one is incorrect in each sequence? Pause this video.

Spot them.

Question 2, part c, same skill.

Use the graph to spot the incorrect term in that arithmetic sequence, starting 1/2, 7/12, 2/3.

For question 3, Laura is asked if the number 117 is in the sequence starting negative 8, negative 2, 4, 10, 16? She says, I'll draw a graph and see if 117 is in line." Write a sentence explaining why this might not be the most efficient method to use on this occasion.

Feedback now.

Sam's asked to continue this arithmetic sequence and plot it.

They write the blow values and plot them, and I asked you to write a sentence about why you know there's been an error.

You should have written something along the lines of, arithmetic sequences plus in a straight line, and on the seventh term, the line shifts.

So there must have been an error here.

It must be the seventh term, 178,250.

That is no longer in line with that arithmetic sequence.

The graph shows you instantly there's been an error.

You might also have noticed in the numbers, they ended 850, 350, 850, 350, and then at that point there was a shift to 250.

It was a calculator misread on Sam's behalf.

The seventh term was actually 172,850.

You can see how that's been misread, miswritten.

When we look back at the numbers in the sequence, we might have spotted that one.

Question 2.

You can clearly see it's that term that's out of line.

Which one's that, the first, second, third, fourth? The fourth term is not in line.

So 0.

3 must not be its value.

For part B, perfectly aligned until we get to there.

One, two, three, four, five, six, seven, eight, that's the eighth term that's not in line.

So 0.

35 must not be its value.

For part C of question 2, we are in line until we hit that moment.

One, two, three, four, five, six, that's the sixth term.

So 9/15 must not be its value.

For third question, Laura was going to draw a graph and see if 117 was in line with the other points.

You might have included in your sentence, "The term 117 will have quite a large term number.

If it is in this sequence.

It would take a long time to plot this without technology.

It'll be more appropriate and certainly quicker to use that aren't even features of this sequence to justify why 117 is not in this sequence." Graphing sequences is awesome, but it would take a while to plot this sequence all up to 217.

So just saying 4, 10, 16, even numbers, even common difference.

That sequence will always remain even.

Whereas 117, is a odd number.

It cannot be in the sequence.

We could have solved this one far quicker for Laura.

Finally today, using the nth term.

So beyond graphing with technology, identifying features, there's many other methods we could use to see if large numbers are terms in a sequence.

Is 607 in the arithmetic sequence starting 1, 7, 13, 19? Well, you know the N term, it's 6n minus 5 and we can approximate where 607 is because it's going to be near the 100th term.

Six lots of 100 make 600.

So 607 is gonna be somewhere near the 100th term.

We find that 100th term by substituting into the nth term expressions.

Six lots of 100 minus 5 gives us 595.

From there we can count, oh, we know it's an arithmetic sequence with a constant difference of positive 6.

So 595, 601, 607.

It's the 102nd term.

So is negative 607 in the arithmetic sequence starting 2, negative 1, negative 4.

Well the nth term is 5 minus 3n.

It's gonna be somewhere near the 200th term.

So I'm gonna substitute in n equals 200.

5 minus 3 lots, 200 gives me negative 595.

Negative 595, moving in steps of negative 3, the n coefficient is negative 3.

So this is decreasing by 3 each time, minus 595, negative 595, negative 598, negative 601, negative 604, negative 607.

That's a 200th term, the 201st term, 202nd, 203rd, 204th.

Yes, negative 607 is in the sequence.

It's the 204th term.

Your turn now.

Is negative 58 in this arithmetic sequence.

Pause this video and work that one out for me.

So the nth term, you should have found to be 860 minus 9n.

Approximately near the 100th term.

When you find that 100th term, you find it to be negative 40, and then you count on in steps of negative 9, negative 40, negative 49, negative 58, there it is.

It wasn't the 100th term or the 101st, it was the 102nd term.

Lovely work.

Next, what if that large number is not in our sequence? What language do we use to justify the fact that it's not.

Is 841 in the arithmetic sequence starting 3, 7, 11, et cetera? Well, the nth term is 4n minus 1.

841 is gonna be approximately the 210th term.

210th term is 839.

From there, we can count on, 839, 843, 847 is how the sequence is gonna go.

It bypasses 841.

So we know 841 is not in the sequence.

We would use the language.

839 is the 210th term, 843 is the 211th term, therefore 841 cannot be in the sequence.

Just check you've got that.

Which best justifies that 295 is not in the arithmetic sequence starting negative 4, negative 1, 2, 5 with the nth term 3n minus 7? Is it n equals 100, 3 lots of 100 minus 7 equals 293.

So it's not 290, 293, 296, 299, didn't include 295.

Is that the best justification? Or is it 3 lots of 100 minus 7 equals 293, 3 lots of 101 minus 7 is 296, 100th term is 293, 101st is 296, so 295 cannot be in the sequence, which is the best justification.

Well done.

The third one.

The 100th term is 293, the 101st term 296, and 295 cannot be in the sequence.

Whilst we work out the 100th term and find it at 293, you know the next term's gonna miss 295.

But you can't just leave it as that because you haven't justified that the next term is beyond 295.

You'd need to work out the 101st term and show that it's not 295, it's beyond 295 if you want to use that justification.

In part B, we haven't made any justification as to are those numbers in this sequence.

You need to justify that.

That's why part C was by far the best justification.

What other language could be used to justify a non-term? So same question, 841 is it in that arithmetic sequence, 3, 7, 11, et cetera, nth term 4n minus 1.

And use substitution in our calculators.

So we knew it was near the 210th term.

That's how I typed 4n minus 1 into my calculator.

Now I'm gonna use the left arrow key to change my 210th term to my 211th term.

So I've got the terms 839, 843.

I can now see that it's not in the sequence.

If I wanted to make 841 with 4n minus 1, I need to substitute in 210.

5.

I could play with my calculator, find that number.

I need an n value of 210.

5.

But we can't have that because N has to be a whole number.

There is no 210.

5th term in a sequence.

So we could use the language, there's no whole number we can substitute into 4n minus 1 to make 841, and n, the term number, needs to be whole, therefore it cannot be in the sequence.

Just check you've got that now.

What does this calculated display show us.

That 911 is in the sequence, 10n minus 6.

There's no whole number.

We can substitute into 10n minus 6 to make 911.

Therefore, it's not in the sequence.

Or 911 is the 91.

7 term in the sequence 10n minus 6.

Well done.

It was B.

There's no whole number we can substitute in, in order to make a value 911.

Therefore, it's not in the sequence.

Term numbers have to be whole.

There's no 91.

7th term.

Practise time now.

Question one, use nth term and approximation to find 400 in the arithmetic sequence starting 10, 23, 36, 49.

Question 2 is 400 in that arithmetic sequence? I'd like you to write a sentence to justify your answer.

Same for question 3, is negative 12.

75 in that arithmetic sequence? Again, write a sentence or sentences to fully justify your answer.

Pause this video.

Give those three a go.

Feedback time.

The nth term and approximation to find 400, right? The nth term is 13n minus 3.

We can approximate that 400 somewhere near the 30th term.

When we substitute n equals 30 in 13 lots of 30 minus 3, that gives us 387.

We're incredibly close to 400.

If we count on in those steps of 13, because the arithmetic sequence moving in a constant difference of positive 13, 387, 400, there it is.

It's the 31st term.

We could say 400 is in the sequence.

It's the 31st term.

Question 2, the nth term is 8n minus 3.

400 is approximately the 50th term.

The 50th term is 370, from there we can count on.

And you can see we miss 400.

So how do we then justify that 400 is not in the sequence.

We'd say 394 was the 53rd term, 402 was the 54th term.

Therefore, 400 cannot beaten the sequence.

We could have done it slightly differently, that justification.

We know the nth term is 8n minus 30.

That is using my calculator to show that when I substitute in n equals 53, we land on 394, n equals 54, we land on 402.

If I were to make 400, I would need to substitute in an n value of 53 3/4 or 53.

75.

So we could say, there is no whole number.

I can substitute into 8n minus 30 to make 400.

And n, the term number, needs to be whole, therefore it is not in the sequence.

You would of course write down that working out, out of your calculated display along with that sentence.

Question 3, the nth term 21.

6 minus 0.

35n.

21 minus 35 equals negative 14.

So it's somewhere near the 10th term as in 100 lots of 0.

35 makes 35.

I can approximate it to be somewhere near the 100th term.

So the 100th term is negative 13.

4.

We count back from there because at negative 13.

4, we've actually gone past negative 12.

75 in this sequence.

So when we count back, we find that's the 100th term, the 99th term, the 98th term, and we could say negative 12.

7 is the 98th term negative 13.

05 is the 99th term.

Therefore negative 12.

75 cannot be in the sequence, right? We could have done that with our calculator, but if I said take the nth term expression and make the exact value of negative 12.

75, it might have taken you a while to spot that you had to type in an n value of 98 and 1/7.

So it would've been less appropriate to use that method on this occasion.

Sadly, I send this today's lesson and I say sadly, because I've really enjoyed it.

So in summary, I can determine whether a number is a term of a given arithmetic sequence in a variety of ways by looking at the feature of the sequence, by graphing it and also by using the nth term.

Do join me again soon for more mathematics.

Bye for now.