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Thank you for joining me for today's lesson.

My name is Ms. Davis, and I'm gonna be guiding you as you explore some of these new and exciting sequences that we are looking at today.

Make sure that you've got everything you need before you start watching this video.

It's always a good idea to have a pen and paper so that you can jot things down and explore things in your own time.

Let's get started then.

Welcome to this lesson, where we're gonna be problem solving with further sequences.

So we're gonna be using all of the knowledge we have about sequences to solve some problems. We're gonna start today's lesson by talking about interest.

Now, interest is money added to savings or loans, and compound interest is calculated on the original amount and the interest accumulated over the previous period.

Some geometric sequences might come up today.

A geometric sequence is a sequence with a constant multiplicative relationship between successive terms. In a geometric sequence, the constant multiplier between successive terms is called the common ratio.

There's also other types of sequences that might appear in this lesson.

So let's start by looking at compound interest.

Izzy is investigating different types of loans.

She says, "I've used an online loan calculator for one company.

Somebody who borrows 80 pounds for a year then owes 120 pounds.

But if they borrow the money for two years, they would owe 180 pounds.

Andeep says, "That's interesting.

The amount owed increased by 40 pound, then by 60 pound." Why do you think that happened? Pause the video and have a bit of a discussion.

What's going on here? Right, well loans gather interest the longer they are left unpaid.

If the interest is compound interest, then the amount of interest will increase each period.

Make a prediction of how much will be owed if borrowing 80 pounds for three years.

Use the table to help you.

Okay, keep hold of your predictions.

Andeep says, "It increased by 40 pounds than 60 pounds, so for another year I think it'll be 80 pound more." What rule do you think Andeep is using to make his prediction? It seems like he thinks the amount of interest is gonna increase by 20 pound per year.

So increase by 40, then 60, then by 80.

Oh, Izzy says, "Actually it was 270 pound, so an increase of 90." Can you see a rule linking these values? Have a think.

If you can't spot it, don't worry.

We'll look at it together.

Right, well, the amount owed each year is following a pattern.

If we write those values as a sequence, we can explore this further.

It's not a linear sequence, as the amount added on, which is the interest, is increasing each year.

It's not a quadratic sequence with common second difference of 20, 'cause Andeep just tried that.

What other type of sequence could it be? I wonder if you thought geometric.

How could we test to see if it could be geometric? Well, we need to divide successive terms to see if there's a common ratio.

We do 120 divided by 80.

That's 1.

180 divided by 120.

270 divided by 180.

So, so far this could be forming a geometric sequence.

Another company says they charge an interest rate of 10% per annum.

Let's see what borrowing 80 pounds would look like here.

So borrow 80 after one year at 10%, that'd be another eight, so 88.

And 10% of 88 is 8.

So if we add that on, we get 96.

Andeep says there's an easier way to increase by 10%.

We can multiply by 1.

1, and this way of increasing by percentage using a multiplier is gonna be really important in today's lesson.

So multiply to increase by 10% is 100% plus 10%, which is 110% or 1.

So 96.

8 times 1.

1 is 106.

48.

So let's see if this one also forms a geometric sequence.

Well, we've just said that increasing by 10% we can multiply by 1.

So yes, we're gonna get a geometric sequence with a common multiplier of 1.

As long as the interest rate remains the same and no money is repaid, the value of a loan over time will form a geometric sequence, as with compound interest.

So there's our first one.

What was the interest rate for the first company Izzy investigated? Can you find that out from the information we have? Yeah, 1.

5 is the multiplier to increase by 50%.

So when we multiply by 1.

5 each time, our values are increasing by 50% each time.

Izzy says, "This was a payday loans company, which is probably why the yearly interest rate was so high." And that's true.

If you are borrowing from a company that's only normally gonna lend you money for a couple of days, then their interest rates are gonna be really high if you looked at them yearly.

If you're looking at something like a mortgage, you're gonna get interest rates at an awful lot lower.

The second company charged 10% interest, which is why the common ratio is 1.

Right, I'd like to match the percentage change to the corresponding multiplier.

Off you go.

Right, the first one was the trickiest.

2% increase would be 1.

012.

2% would be 1.

02.

12%, 1.

12.

And 20%, 1.

Okay, what is the common ratio for this geometric sequence? Off you go.

The common ratio is 1.

25.

We divide successive terms. You can see we always get 1.

25.

If you're doing this without a calculator, the easiest terms to look at is 125 and 100.

125 divided by a hundred we can do in our heads.

That's 1.

25.

If this was a multiplier for an interest rate, the interest rate would be 25%.

So we can use this fact that money owed after compound interest forms a geometric sequence to solve problems. A loan company charges 2% compound interest a month.

If a 1000 pounds was borrowed, how much will be owed after five months? Let's do that together.

We start with 1000, and that's the start amount, not the amount after one month.

After one month, we can multiply by 1.

02.

1020.

Then we do the same and keep going until we get to the fifth month.

1,104 and eight pence.

Now, if you wanted a more efficient way, you could do 1000 multiplied by 1.

02 to the power of five, 'cause we're multiplying by 1.

02 five times.

Right, somebody else is borrowing from the same company, and they owe 663 after one month.

How can we work out how much they borrowed? What do you think? Right, so we know that after one month we had 663 pounds, and we know to get the next month we multiply by 1.

02.

So to get a previous term, we need to divide by 1.

02.

663 divided by 1.

02 is 650.

Right, Izzy finds a loan calculator for another company.

"I forgot the start value, but after four years, 5,000 pounds would be owed, and after five years, 6,250 would be owed." How could we then find the interest rate? What do you think? These are just successive terms in a geometric sequence.

So we need to divide them to find the common ratio.

If I put them in the table, it'll help me get this the right way around.

If I do 6,250 divided by 5,000.

And that gives me 1.

25.

The interest rate then is 25% a year.

How could we find the start value? What do you think? All right, we can divide by 1.

25 until we get back to the loan amount.

We get a loan amount of 2,048.

Quick check.

The amount owed on a loan is 2000 pounds after five years and 2,600 pounds after six years.

What is the interest rate as a percentage? Well, if you said 30%.

We divide successive terms, so 2,600 divided by 2000, which is 1.

3 is the multiplier to increase by 30%.

Some of our sequencing skills will help us solve trickier problems. After two years, a loan is worth 1000, but after four years, that rises to 1,210.

What is the interest rate on this loan? We'll do this together.

So I'm gonna put the terms in the table.

So I know the second year is 1000, and the fourth year is 1,210.

I need to know what the overall multiplier is, and then I can work out the multiplier per year.

So the overall multiplier is 1.

21.

I can divide 1,210 by 1000.

Now I want the individual multiplier per year.

If we need to multiply by a value twice to get 1.

21, this means we need the square root of 1.

21, which is 1.

And you can check that with your calculator if you wish.

1000 times 1.

1 is 1,100, times 1.

1 is 1,210.

Now we need to remember to answer the question 'cause the question asked, "What is the interest rate on this loan?" The interest rate is 10%.

1 is the multiplier to increase by 10%.

Right, time for practise.

For question one, 6,250 pound is borrowed at a compound interest rate of 4% per month.

Can you answer those four questions? For question two, we've got a value of a loan over four years if no repayments are made.

Use the table to work out the compound interest rate for that loan.

For question three, you've got a different loan.

I'd like you to work out the compound interest rate and then how much was borrowed to start with.

Question four.

A loan gathers compound interest yearly.

If no repayments are made, 6,820 is owed after two years, and 24,389 would be owed after three years.

You could use the table at the bottom to help you.

You've got three questions to answer.

Question five.

A loan model for borrowing money to buy a washing machine is given below.

Andeep says, "I don't think this loan charges compound interest." Read the information.

Is Andeep correct? Make sure you explain your answer.

And then for our final challenge.

1000 pounds was borrowed from a company charging yearly compound interest.

After two years, 1,440 was owed.

What's the interest per year? Then you've got a similar question for question seven.

And then for question eight, 3000 pounds was borrowed from a company charging yearly compound interest.

After three years, 3,993 was owed.

What is the interest rate per year? Give those a go.

Let's have a look then.

So you should have 6,500 after one month.

The multiplier is 1.

04.

No repayments are made.

You do 6,250 times 1.

04 cubed, which is 7,030 and 40 pounds.

How much would be owed after one year? Well that's 12 months.

So you get 6,250 times 1.

04 to the power of 12, which is 10,006 pounds, 45 pence.

For question two, if we divide 1,300 by a thousand, we get 1.

That's 30% compound interest each year.

Do the same for the next one.

We get 1.

02.

That's 2% compound interest each year.

If we divide the amount after one year by 1.

02, we get the original amount, which is 125,000.

Question four, we divide the terms, we get the interest rate.

That's 1.

45 as a multiplier.

So 45%.

We've got the amount after three years, so if we want the amount after five years, we just need to times by 1.

45 twice.

We get 51,277 pounds and 87 pence.

How much is borrowed to begin with? Well, we need to divide by 1.

45 twice or divide by 1.

45 squared, which is 8,000.

Question five.

Andeep is correct.

The values seem to be forming a linear sequence with three pounds added each month.

This suggests that the interest is the same each month.

This will be simple interest, not compound interest.

And our challenges.

So dividing the terms we know gives us 1.

44.

We need to square root to get a single multiplier of 1.

2, which is 20% compound interest each year.

Doing exactly the same for seven.

Get a multiplier of 1.

1664.

And it's important you use that exact value in your square root, which is 1.

08.

8% compound interest each year.

Oh, well done if you've got this one.

So once we found our overall multiplier, we need to find the cube root, and that gives us 1.

1, 10% compound interest each year.

Right, now we're gonna explore some patterns formed by Pascal's triangle.

You don't need to know what that is.

We'll talk about it now.

So we're gonna see if we can identify certain sequences in a mathematical diagram called Pascal's triangle.

Here are the first five rows.

There's row one, two, three, four, five.

Can you work out how each row of the triangle is generated? Pause the video.

What do you notice? Did you spot that each number is the sum of the two numbers above it? For example, three is one add two.

What would row six be? Pause the video, work it out.

Right, we're gonna have 1, 5, 10, 10, 5, and 1 again.

Quick check.

What are the missing numbers in row seven of Pascal's triangle? And then we're gonna use this.

The outside number is always one.

Then we've got one add five is six, five add 10 is 15, 10 add 10 is 20.

So Pascal's triangle is fascinating.

It's used for different branches of mathematics, but it contains many sequences and patterns, and that's what we are interested in today.

Laura says, "It is symmetrical!" What do you think she means by this? Right, the numbers either side of the middle are the same but in reverse order.

And noticing this makes it easier to work out what the numbers are in each row.

As soon as you get to the middle, you know you can just repeat the same as on the left hand side.

We can also look at patterns in diagonally adjacent tiles.

I'll show you what I mean on the diagram.

We can see that the outside diagonal column, I'm gonna call that a diagonal column, only contains the number one.

It's the same for the other side.

Can you find a diagonal column which forms a linear sequence? Pause the video.

It's the second diagonal column.

It forms the linear sequence which starts one, two, three, four, and they're the positive integers in order.

You might have said it's the sequence with nth term, n.

Well, I don't know if you remembered that.

The second diagonal column from the left is identical because of this idea of symmetry.

Right, real challenge now.

Can you find numbers which form a quadratic sequence? That's a sequence with a common second difference.

See if you can spot it.

It's that third diagonal column.

You can look at it from either side.

1, 3, 6, 10, 15.

You can see the numbers are increasing by two and three, then four, then five.

They have a common second difference of one.

Do you notice anything else about these numbers? They are the triangular numbers.

Laura says, "I have shaded these numbers to make a pattern." Why do you think Laura has chosen these numbers to shade? What rule can you spot? You might have said they're all even numbers, or they're all multiples of two.

Jacob is shading non-adjacent diagonals in different colours.

You'll see what I mean by this when I shade them.

So there's the first one.

The second one.

That's the next diagonal.

And so on.

"If I add up the numbers on the non-adjacent diagonals, I can spot a pattern," says Jacob.

So we've got 1, 1, 2, 3, 5, 8, 13.

What sequence do you think he's noticed? Did you spot that it's the Fibonacci sequence? One add one is two, one add two is three, two add three is five, and so on.

Using the Fibonacci sequence, what should the sum of the next diagonal be? Should be eight plus 13, which is 21.

Will this be the case? How can you tell? Yeah, it will.

The diagonal tiles we already have sum to 20.

Well, we know the outside one is one.

So that'll give us 21.

Time for you to have a play around with Pascal's triangle.

Here are the first 13 rows.

To start with, I'd like you to shade all the odd numbers.

Do you notice anything about how they're arranged? Then you've got two questions to answer once you're done.

Off you go.

Laura says, "The third diagonal column from the left is the triangular numbers." We looked at this with our smaller version before.

Can you show that Laura is correct for the diagram so far? And then shade all the triangular numbers.

It's easy to miss a few of the triangular numbers.

I'm gonna give you a hint with this one.

The largest triangular number that's in this diagram is 210.

See if you can get all those triangular numbers.

Off you go.

Jacob has noticed a pattern in the sum of each row.

Have a look at the diagram.

What pattern do you think Jacob has noticed? Then I want you to test whether that continues for the next two rows and tell me what you think the total of row eight will be.

I've marked row eight on the diagram.

Off you go.

And question four.

This time I'd like you to shade all the multiples of three.

Do you notice anything about how they are arranged? Something that might help you is the divisibility test for multiples of three.

If the digits sum to a multiple of three, then the number is a multiple of three.

That's gonna help you with some of the larger ones.

And finally, you've got complete freedom as to what you want to shade this time.

Here are the first 16 rows of Pascal's triangle.

So you've got a little bit more.

You could choose a new type of number to shade, so maybe a different multiple or a different type of sequence that you can think of.

Or you could pick one of the previous patterns and continue it for 16 rows and see if it carries on following that pattern.

Once you've created an amazing pattern, come back, and we'll look at our answers.

There we go.

That is the pattern that you should have.

It seemed to form a sort of triangular style pattern.

The row that contains only odd numbers is row eight.

Now you might have made a prediction about whether there'd be another row containing only odd numbers.

Actually, row 16 contains only odd numbers, and we can look at that in our larger version in question five.

Might wanna make a conjecture now about which rows will only contain odd numbers.

I wonder if it carries on with any kind of pattern.

And here are our triangular numbers.

So we can check Laura's correct.

We've definitely got one then three then six then 10, which are the first four triangular numbers.

And we're following this pattern with a common second difference of one.

We're adding two, then three, then four, then five, and so on.

So yes, we have got the triangular numbers in that third diagonal row.

Now, well done if you've got all of them.

Hopefully got the two diagonals shaded.

But then, six is a triangular number, so we need to make sure that we've shaded that.

10 is a triangle number, so we need to make sure we've shaded them as well.

And then the trickiest bit was that 120 is a triangular number, and I told you that 210 was as well.

We get an interesting kind of arrowhead shape in this diagram.

Right, Jacob's patterns.

I wonder if you said the sum of the rows double each time.

So we've got one, then two, then four, then eight.

We're doubling them each time.

They form a geometric sequence.

Well, I don't know if you used that language.

You might have said it's the powers of two.

We've got two to the power zero, then two to the one, and two to the two, and so on.

For the next two rows then, if this continued, we'd have a total of 16 and a total of 32.

And that does happen.

What would the total of row eight be then? Well, if we multiply by two again, we get 64, multiply two again, we get 128.

So row eight should have a total of 128.

And that is true.

So we seem to have a geometric sequence with the totals of the rows.

The multiples of three.

These were quite fun.

Again, they formed a different kind of triangular pattern, and that sort of makes sense because if two numbers are a multiple of three, then their sum is gonna be a multiple of three.

So that's gonna make an upside down triangle shape.

And then you've got this big upside down triangle in the middle.

I wonder if that pattern will continue as we carry on.

And then you could have chosen any sequence to play about with here.

I decide to show you the rest of the odd number pattern that we started with.

That's quite a nice pattern.

And what you end up with is these smaller triangles inside a bigger triangle.

I hope you enjoyed playing around with Pascal's triangle and spotting all these sequences.

Right, you've worked really hard today, as well as having a little bit of fun.

Thank you for joining us.

I look forward to seeing you in a future lesson.

I've finished the video