Lesson video

Hello, Mr. Robson here.

You are wise, you've chosen maths today, and it's a wonderful choice, because today we're talking about sequences, and I love sequences.

Our learning outcome is that we'll be able to appreciate that any term in an arithmetic sequence can be expressed in terms of its position in the sequence.

Key words for today, the key word is nth term.

Now that might be new to you and it might sound unusual to you.

The definition, the nth term of a sequence is the position of a term in a sequence where n stands for the term number.

For example, if n = 10, this means the 10th term in the sequence, nth, termed n, will be referenced throughout this lesson and you'll get a clear understanding of what this means.

2 parts to the lesson, we'll start by talking about the position of a term in a sequence, and then we'll move on to looking at the general term of a sequence.

And you know that we mathematicians love to generalise, but let's begin by talking about the position of a term in a sequence.

Our Oak students are discussing expressions.

Expressions are really important in algebra, in mathematics.

So how do you express the perimeter of any rectangle? We could express the lengths of any rectangle is base and height.

So how do you express the perimeter? Sam's gone for b + b + h + h.

Jacob's gone for 2b + 2h, Andeep's gone for 2(b + h).

Question for you now, who is right? Pause this video, tell the person next to you or say it back to me on the screen.

The good news is all of our Oak students are correct.

All of these expressions can be used to express the perimeter of any rectangle.

However, Jacob and Andeep's expressions are simplified Andeep's looking at it in terms of 2 lots of b and h.

If I look at Sam's expression, I can see 2 lots of b and h.

Jacob's looking at it as 2 bs and 2 hs.

If I look at Sam's expression, I can see 2 bs and 2 hs, but Jacob and Andeep's expressions are written more concisely, more compactly, more simply, and we mathematicians like to write things as simply as we possibly can.

In today's lesson, you are gonna see that sequences are no different.

We want to express those as simply as possible too.

Can you see a pattern in this sequence? 2, 4, 6, 8, 10, just one second, say something back to me.

There were lots of things you could have said.

The term-to-term rule is +2, it's our 2 times table you might have said, you could have said it starts at 2 and counts up in 2s.

Lots of things you could have said.

Andeep goes on to say, "The 100th term in this sequence is 200." Is he right, and if so, why? Pause this video, tell the person next to you or speak aloud to me on the screen.

He is right, because the term value is always twice the term number in the 2 times table.

The term value is always twice the term number in the 2 times table.

What does that mean? It's easy to think about it if we put this into a table, term number, the 1st term, the 2nd term, the 3rd term, the 4th term, the 5th term, and the term values 2, 4, 6, 8, 10, IE, the third thing in our 2 times table is 6.

And you get that relationship of multiply by 2 each time.

So 1 x 2 makes 2, 2 x 2 makes 4, 3 x 2 makes 6.

That's how our 2 times table is constructed.

So if we went along in that table all the way to the 100th term number, we would take 100 and multiply it by 2 and we would get 200, so Andeep would be correct.

5, 10, 15, 20, 25.

What relationship helps us find any term number in this sequence? I think you spotted a link between this sequence and the last sequence.

It's a times table.

Alex says that the link is, "Multiply the term number by 5," multiply the term number by 5.

Well that works.

If we put in a table, again the first term multiply by 5, we get a term value of 5.

The second term, multiply 2 by 5, we get 10 and so on.

It's always your term number multiplied by 5, giving your term value.

When we're talking about the sequence that is our 5 times table, and it'll work for any term in the sequence.

It will work for any term in the sequence.

For example, the 10th term, we would do 10 multiplied by 5 and we'd get 50.

The 10th number in your 5 times table is 50.

Whilst the language is precise, and by language I mean what Alex is saying, "Multiply the term number by 5," that language is precise, but it's not very efficient and we mathematicians like to be efficient.

To improve the efficiency here we say n to mean any term number.

We say n to mean any term number.

Could you just say that sentence back to me again? Lovely.

We say n to mean any term number.

So our table is going to take a slightly different presentation now, just to remind you the definition I read to you a moment ago, "The nth term of a sequence is the position of a term in a sequence where n stands for the term number.

N = 10 means the 10th number in the sequence." So instead of saying term number, we're gonna say term number (n).

Once we do this, Alex can now say, "Multiply n by 5." Using n the language has become more concise and we like concise, efficient, simple things in mathematics.

Quick little check now, in sequences we use the letter n to represent what? Is it the term number, the term value or the relationship between term number and term value.

Pause this video.

Say your answer to the person next to you or say it to me on screen.

It's a, the term number, the letter n represents our term number in sequences.

Another check, in sequences, n = 5 means what? Does it mean a, the solution to the equation is 5, b, the value of the term is 5, or c, the 5th term number? Pause this video, tell the person next to you or tell me aloud at the screen.

The answer's c.

It's the 5th term number, n = 5 means the 5th term number.

In our learning, we're gonna go on to look at relationships between n and term value.

The term value will be determined by that relationship, but n = 5 means the 5th term number.

Another check.

When discussing the 100th term number, which would be used? Is it a, 100, b, n + 100, or c, n = 100? Pause this video.

Say your answer to your person next to you or say it aloud to screen.

It's c, n = 100.

100 is just a number, n + 100 is an expression, n = 100 means the 100th number in a sequence.

Next, Aisha and Jun are laying out tables in the dining room and they notice a pattern.

Hmm, those are tables, and the circles are people sat around it.

Jun notices 8, 10, 12, 14.

The number of people makes an arithmetic sequence, is Jun correct? Pause this video and have a little think.

Yes, a common difference of 2 each time makes it arithmetic, and if you drew the next set of tables and people, you would find 16 people, it will keep going up by +2 every time.

Aisha says, "I have a rule! The number of tables x2, add the 6 on the sides = the number of people." Does her rule work? Pause this video and test that rule.

It does work.

It's an awesome rule.

For the first table, take 1, multiply it by 2, add 6, we get 8.

That's the 8 people around that one table.

When we have two tables, take 2, multiply it by 2, add 6, we get 10 people.

There's 10 people around our two tables.

For the third pattern, three tables, multiply that by 2, add 6, we get 12, and you'll see 12 people around those tables, for the fourth, multiply by 2, add 6, we get 14.

That's the 14 people around those tables.

Aisha's rule works wonderfully, but we can make it a little more concise.

Aisha and Jun put their results in a table, pardon the pun.

Table number (n) 1, 2, 3, 4, number of people, 8, 10, 12, 14.

They put their results in a table.

Aisha now says, "If n represents the number of tables my rule is easier to say, n x 2 + 6 = the number of people." That's a much easier expression to work with, n x 2 + 6, n being the table number.

Jun congratulate her, "Your rule works Aisha! When n = 4 we have (4) x 2 + 6 = 14 people." Your rule works, but, "What if," I wonder where this is going, "I use n = -2, we'll have (-2) x 2 + 6 people." (-2) x 2 + 6 = 2 people.

So when n equals -2, we'll have 2 people.

What's wrong with Jun's idea here? Pause this video.

Explain what's wrong with the person next to you or say it back to me on screen.

The problem is, "This would mean -2 tables, <v ->2, -1 or zero tables won't exist.

</v> So n has to be a positive integer in this case." We start with 8 people, then 10 people, then 12 people.

We start with 1 table, then 2 tables, then three tables.

We don't in this sequence have zero tables or -1 tables or -2 tables.

So in this case, n has to be a positive integer.

Let's just check you've got that, n, the term number can be zero.

Is that true or is it false? I'd like you to justify your answer with either, because any number multiplied by zero is zero, or because the first term in a sequence is the first term, there is nothing before it.

Pause this video, declare it true or false, and justify your answer.

It's false, because the first term in the sequence is the first term.

There's nothing before it in this sequence.

5 is the first number in this sequence, there's nothing before it.

Practise time now.

Question 1.

Match the term number (n) to the term values (T).

One has been done for you.

In part a, we see the term numbers 1, 2, 3, 4, 5, and the term values 2, 5, 8, 11, 14, and I've matched for you that when n = 1, T = 2, IE, the first term number has a value of 2.

Can you match up the second term number, the third term number, the fifth term number to their respective term values, and then do the same for table b once you are finished.

Pause this video and try that question.

For question 2, Andeep and Jacob are looking at an arithmetic sequence.

The sequence goes 45, 40, 35, 30 25.

You can see those term values in that table.

Andeep says, "I see the pattern! When n equals 0, the term is 50." Jacob says, "I agree! Next would be when n = -1 the term will be 55." Write a sentence explaining why Andeep and Jacob are wrong.

Pause this video and write them a sentence.

Feedback.

Matching term numbers (n) to term values (T).

The first one was done for us on part a.

The first term number had a value of 2.

The second term number where n = 2 has a value of 5.

So we match up n = 2 to T = 5.

The third term number where n = 3 has a value of 8, and the fifth term number has a value of 14.

So we match n = 5 to T = 14.

In part b, we had a decreasing sequence.

5, 4, 3, 2, 1.

The first term number had a value of 5.

So we matched n = 1 to T = 5.

When n = 2 or at the second term number that's got a value of 4.

When n = 3, the third term number had a value of 3.

When n = 4, the fourth term number had a value of 2.

When n = 5, the fifth term number had a value of 1.

Next, Andeep and Jacob, and we had to write a sentence to explain to them why they couldn't do what they were doing.

You might have written: "There is no zero term, or -1 term in this sequence.

The first term 45, where n = 1, is the first number in this sequence.

There is nothing before it." Onto our second learning cycle now, the general term of a sequence.

5, 10, 15, 20, 25, we've seen that before and we've seen this table before.

Only I've left some blank spaces here to help explain something.

This is the 5 times table, where we take the term number and multiply it by 5 to get the term value, IE, the third number is 3 x 5, 15.

The third number in our 5 times table is 15.

We have this relationship between the term number and the term value.

What's the 10th term number in this sequence? Yeah, 10 x 5, 50.

What's the 25th number in the sequence? That was pretty sharp, yeah, 125.

What's the 100th number in the sequence? 500.

What's the nth number in the sequence? Well done if you said n x 5, taking the term number and multiplying it by 5 will give us the term value in this case in our 5 times table, this enables us to write a general term for this sequence.

If I said give me the nth number in that sequence, you would say, well that's n multiplied by 5, and you've got an expression for any value in that sequence, which will work every time.

Using n, we are able to write a general rule for the sequence, and we can write a general rule for any arithmetic sequence using n.

Quick check now, what would a general rule for this sequence be? This is the sequence 7, 14, 21, 28, 35, which should be familiar to you.

What would the general rule for this sequence be, n = 7, n + 7, or n x 7? Pause this video, tell the person next to you or say it back to me on screen.

It's n x 7.

You see the relationship in the table there between the term number and the term value.

The first term has a value of 7.

The second term has a value of 2 x 7, 14, and so on.

So we take n and multiply by 7 and we can write a general rule of n x 7 for this sequence.

So using n, we can write a general rule for any arithmetic sequence.

That doesn't look like an arithmetic sequence, but is it? It is if we start to count something like the number of green squares.

So if I said pattern number 1, how many green squares? You'd say 4.

Pattern number 2, how many green squares? You'd say 8.

Pattern number three.

You can count those 12 green squares.

When we start to spot a pattern there, you can see the green squares are going up by 4 every time.

When we move to the fourth pattern, if you drew that, you'd see that there's 4 more green squares.

We've got 16 green squares in total.

The fifth pattern, there'd be 4 more green squares, we'll have 20 green squares in total.

That pattern is going to go on, but if we can find a general term, a general rule, then we don't need to draw pattern 10.

We don't need to draw pattern 25.

We don't need to draw pattern 100, and I really wouldn't fancy drawing pattern 100 in this case.

So can you fill in the rest of this table, and can you write a general rule for the number of green squares? Pause this video and give it a go.

So you notice that there's 4 green squares for every pattern.

So pattern 10 must be 10 x 4 to get 40 green squares, pattern 25 must be 100, 25 x 4.

100 x 4 gives us 400 green squares so what would the general rule be? n x 4 will tell us any value we wish in this sequence, Right, same pattern, but to complicate things ever so slightly, we're now gonna say all squares, not just the green squares, all squares.

If I give you the first few terms there, that now means 5, 9 and 13, as in in the first pattern there's 5 squares.

In the second pattern, there's 9 squares.

In the third pattern, there's 13 squares.

What's the general rule for this sequence? Jun's given me an idea, "4 times the pattern number plus the middle 1." Okay, Jacob's given us another idea, "Pattern number x4, plus 1.

Then Aisha has given us, "n x 4 + 1." So do any of our Oak students have a general rule for this sequence? Pause this video and have a think about what the three of them have proposed.

You should have noticed that all of our students' rules work.

It's Aisha's, which is the most concise, we can say 4 x the pattern number plus the middle 1, as Jun said it.

We can make that a bit more concise by going pattern number x 4 plus 1, as Jacob said it.

But if we use n to represent the pattern number, we have an nth term expression, which is much more concise, n x 4 + 1 works, and it's the simplest way to write this.

Quick check that you've got that.

Match each pattern to its verbal rule and nth term rule.

The left hand side you'll see three different patterns.

Those three patterns can be described by one of those middle sets of words.

Pattern number times 2 plus three makes the number of squares, to which pattern is that attributed? And then once you've linked the words to the patterns, can you link the words to one of those nth term rules? Pattern number times 2 plus 3, which of those concise nth term rules on the right hand side is that? Pause this video, see if you can match those together.

So to start, the top pattern, those L shapes, pattern number times 2 plus 1 gives us the number of squares let's just check that works for the third pattern.

3 times 2 plus 1 is 7 squares I can see 7 squares, super.

So which nth term expression is pattern number times 2 plus 1.

Well that would be the bottom one, n x 2 + 1.

The second pattern has a lovely link here, pattern number times 3 plus 1.

Can you see the times threeness in that pattern? I'm just gonna check it by checking that the second term works, pattern number 2, 2 times 3 plus 1 should have 7 squares.

Pattern number 2 does have 7 squares.

So pattern number times 3 plus 1.

How would I express that if pattern number were represented by n? It would be n x 3 + 1, which leaves for the last pattern, that's pattern number times 2 plus 3 which we would say more concisely as n x 2 + 3.

Practise time now.

Question 1.

Sam spots a general rule for this sequence.

The first pattern number has 4 squares.

The second pattern number has 7 squares.

The third pattern number has 10 squares.

Pattern number x 3 + 1 gives you the number of squares.

This is true, their general rule works, but how would you improve upon it? Like you to write a sentence explaining how you would improve upon it.

For question 2, fill in the table for this sequence and write a general rule for each row.

The top row is pattern number 1, 2, 3, 4, 5, up to n, any pattern number.

The second row is about the shaded squares.

So in the first pattern I can see 2 shaded squares.

How many do you see in the second pattern, in the third pattern, in the fourth pattern? And if we saw 50 shaded squares, which pattern number would that be? The third row is the link of total squares.

For example, the second pattern has 8 total squares.

Is there a link between the pattern number and the total number of squares? For example, if we have 200 total squares, which pattern number is that? And then finally for the perimeter, the perimeter is the distance around the outside of each shape.

So I can see a length of 4, 1, 4, and 1 in that first pattern.

4 + 1 + 4 + 1 is 10, aha, the first pattern number has a perimeter of 10.

Is there a general rule for describing any perimeter? If I had a perimeter of 208, which pattern number would that be? Pause this video.

Give this a go.

Feedback time now.

The general rule works.

Pattern number three, multiply 3 x 3 + 1, we see 10 squares it works, but how would we improve upon it? You might have said use n to represent the nth term, thus writing the rule more concisely.

So Sam's table might have looked like this, pattern number (n), and instead of saying pattern number, we say n, and then the general rule would be n x 3 + 1.

The table, the beautiful thing that it was, you're probably gonna wanna pause just for one minute and just check that your numerical values in the table match my numerical values.

Pause and just check that for me.

So you kind of needed to spot the general rule each time to be able to fill in the numbers in the table because you wanted to see that link between pattern number and shaded squares.

For each pattern number, there were 2 shaded squares.

The first pattern has 2 shaded squares.

The second pattern has 4 shaded squares.

The third pattern has 6 shaded squares.

We always take the pattern number and multiply it by 2 to get the number of shaded squares, which we would express as n x 2.

That's how you knew, when we had 50 shaded squares, we were talking about pattern number 25.

For the total squares, you'd have noticed that was n x 4.

For example, in the fourth pattern we had 16 total squares.

In the third pattern there were 12 total squares.

It was always the pattern number multiplied by four.

And that's what told you that when we had 200 squares, we were talking about pattern number 50.

The last row was the trickiest 'cause the perimeter wasn't, well the shaded squares is our 2 times table, the total squares is our 4 times table.

But the perimeter didn't quite behave as simply.

But you should have noticed a link between the row and the shaded squares and the row perimeter.

2 shaded squares, perimeter of 10, 4 shaded squares, perimeter of 12, 6 shaded squares, perimeter 14.

The perimeter is always 8 more than the number of shaded squares.

We would express that as n x 2 + 8, because n x 2 is the shaded squares.

And if we need to add 8 to that to get to the perimeter, we just do n x 2, add 8.

From working backwards there, you would've known that with a perimeter of 208 we're talking about the hundredth pattern number.

So to summarise today's lesson on expressing an arithmetic sequence, I can appreciate that any term in an arithmetic sequence can be expressed in terms of its position in the sequence.

n represents the term number, n = 10 means the 10th number in the sequence.