Sequences can be linear, quadratic or practical and are based upon real-world situations. If you can find some general rules then it will help you find terms in sequences.
Number sequences are sets of numbers that follow a particular pattern or rule.
If the rule is to add or subtract a number each time, this is called an arithmetic sequence.
If the rule is to multiply or divide by a number each time, then it is called a geometric sequence.
Each number in a sequence is known as a term.
A sequence which increases or decreases by the same amount each time is called a linear sequence.
All right, we are looking at sequences and there are three main kinds of sequences. We've got linear, quadratic and geometric. And we go through all three and then do a couple of exam style questions at the end. So let's look at linear first. Linear is the nicest kind of sequence, it's the simplest one to solve, so hopefully it won't be too bad just to take a step back.
We use N to talk about the position of each number in my sequence. These actual numbers are the numbers we're interested in. This is an actual sequence, we just label them by their position in the sequence. So eleven is my fourth number in the sequence or the fourth term in the sequence. And so we say N equals four for eleven and so on.
So the first thing we're going to do when we're trying to work out a general expression or general rule to define what's going on here is we look at the difference between the numbers and so the difference between first and second numbers is we add on three, right? Two plus three is five, and with five and eight we add on three, eight and eleven. Again, we add on three. And we can see we've got a general pattern for me here. We add on three every time.
And so if we want to have initial guess of the expression that would represent these numbers, can we think of anything that goes up in jumps three? Well, we know hopefully from years and years ago, our three times tables goes up in terms of three. And so my first guess at a representation for this sequence is going to be my three times tables. Well, how do we represent that? Well, my three times tables is just I take one and I times it by three, I get three, I take two, I times it by three and I get six.
I take three, I times it by three and I get nine and so on. This is my three times tables. So every time I want to work out my next number in my sequence, I look at what the value of N is, the position of that number and I times it by three. And so what I'm doing is three N. And like I say, this is like our first guess, basically our first guess at what the sequence could be, what the expression for the sequence could be.
If we look at each term in the sequence for our first term, we've got three, we want two. For our second term we've got six, we want five, we've got nine, then we want eight. We can see we basically just subtract one from every single term in the sequence. And so if I just extend this table down to them, we're going to have a go at a second, better informed guess. Now we're going to say we're going to stick with the uninsured gas of three n, but before we actually do anything we're going to subtract one from it and then we're going to work it out.
And so rather than three, we're going to have two rather than six, we're going to have five rather than nine, we're going to have eight and so on. And what we can see is we're starting to get our sequence come through. Right, we've got two and we've got two, we've got five and five, we've got eight and 811, 1114, 14. And so I know that this is going to be the expression for my sequence. Right let's go on to quadratic sequences.
These are slightly trickier and again we start with the same general principle so we look at the difference between each term. So we've got plus three here, four up to nine. In this case it's going to be plus five, nine to 16 is plus seven. And you probably guess what that pattern is. 1625 is plus nine.
And now the actual thing we do is we look at the difference between the differences and what we can see is that is now a constant value. Plus two. Plus two, plus two. Well if you take a step back for a second we know what these numbers are. We know the rule already for this because these are just square numbers, right?
All I'm doing is I'm taking the value of N, I'm squaring it, I'm taking the value of T, for example, I'm squaring it to get four and I'm not really timing it by anything, I'm just times to get by one. I'm not changing the value after I've squared it. And so what I do is I'm taking N, I'm squaring it and I'm times to get by one. And the reason I'm putting one there is because that's how these two different things are related. We know that our difference between the differences is two and the general rule is we take the difference between the differences, we divide it by T and that then gives us the number in front of X squared and that's the rule that we use to solve quadratic sequences.
Right, moving on to geometric. Next, geometric is quite a bit different from linear and quadratic. I think the easiest way to understand this is we've basically got a starting number and then all other subsequent numbers are kind of calculated from the initial starting number. Geometric just means we've got a common scale factor between the numbers. And we can see here, we've got 26, 18, 54.
So we can see every single time we calculate the next number in the sequence. We're actually times in the previous number by three every single time. Well if we were to lay this out in a slightly different way, hopefully start to make sense how we're going to get a rule or an expression to define the sequence. So I bought my starting number which is two. So I take my starting number of two to get six.
To get the second number in my sequence, I take my starting number and I times it by three to get the next number. To get a third number in my sequence, I take the previous number. In this case, it's six. And again, I times it by three. I'm going to write six.
Instead of writing just six, I'm going to write six as two times three. So I take six, which is two times three, and I times it by three. And that gives me the third number, my sequence, the fourth number in my sequence. To get 54, I take 18, or I take two times three times three and I times it by three.
And so therefore, I take 18 times three and that gives me 54. I can keep doing that. Right. Again, we can see a pattern emerging here. Well, how can I write three times three times three?
Three times three times three is just three to the power. Three times three is just three squared, and three is obviously just kind of three to the one we could think of as three to one. And so whenever I want to work out the next term in my sequence, however many jumps ahead I want to go, I take my starting number and I times it by the scale factor, which in this case is three. So I take my starting number, I times it by the scale factor, and I put that scale factor to the power of however many jumps I need to make to get to that term that I'm interested in working out. So if I wanted to find the fifth term, what I need to do is I take my starting number and I times it by three to the power of how many jumps I need to make.
I need to make 1234 jumps in total. So that'd be three out of four. And we can test that out, right? Two times three to four. Three to the four is going to be nine times 981, and then 81 times two gives me that 160.
So that's part number times the scale factor.
Hopefully that makes sense. All right, let's go on to look at a couple of exam style questions just to finish off this topic. So first of all, we're going to look at arithmetic sequences that's left some random scribbles on here before based on a thing we had on a previous page, right? We've got seven numbers and we've got hopefully a common difference between the two. And if we look at what's happening between six and eleven, we're adding on five.
Between eleven and 16, again, we're adding on five. And we start to see that passing. Imagine, right, what goes up in terms of five. We know that our first guess is going to be our five times tables. So 12345.
If I look at my sequence here, my first guess is my five times tables, right? Five then. So five, then five times one is five. Five times two is going to be ten. Five times three is 15.
It's literally my five times tables, no more complicated than that and where we are to now. So how's our first guess performing? Well, we've got five for our first term, but we want six. We've got ten for our first term, but we want eleven. Then we want 16.
And so, again, I can see that actually, I need to add on one to every single turn. And so that then gives me 611, 1621 and so on. Because I've now kind of recreated this sequence. I know that my expression is going to be five N plus one, and that's going to be my answer. Let's look at quadratic sequences next.
So if we were to look at this one, this is a slightly differently laid out question. So for this one, we're given the expression and we want to actually work out the 10th term of the sequence. Well, under the 10th term of the sequence, n is going to be ten, right? I know if the 10th term N equals ten, that's what N does, right? It tells us the position of the term in the sequence.
And so what I do is I take ten and I show it into this expression. So I put ten into here. So what we end up with is two times ten squared plus four times ten minus one or ten squared is 100 times two. It's going to be 204, times ten is 40 and minus one. So we get two, three, nine, as I was saying.
Right, finally, we'll go on to geometric sequences. So we're giving our start number, which in this case is going to be 100. And we're told that it doubles every 3 hours and we're interested in how many bacteria we have after 24 hours. So how many jumps is that going to be? Well, how many times does three go into 24?
It goes in 24, eight times. And so we're going to have eight jumps. And so, just like we've done the previous page, I've got my start number, which is 100 times my scale factor, when I know it doubles every 3 hours. So my scale factor is going to be two and the power is going to be eight. Well, that's how many jumps we've got.
Well, two times tend to be eight is 256, so it's going to be 100 times two, five, six, which gives me two, five, six, and that is the end of season.
Book a lesson with this tutor
A young, gifted tutor who uses his engineering experience to ignite your curiosity in Maths. A straight 'A' student, he knows exactly how you need to learn to help you ace your exams.