C1

Sequences and Series

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A

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Firstly, what's the difference between a sequence and a series?

A sequence is a list of numbers with an obvious pattern.

2, 4, 6, 8, 10, ... is a sequence.

When I say "obvious" it doesn't have to be that obvious. Here's a less obvious one.

1, 1, 2, 3, 5, 8, 13, ... can you figure out what's happening, or does it look random?

That's right! This is the famous Fibonacci Sequence. To find the 5th term we add the 3rd and 4th. To find the 6th we add the 4th and 5th. etcetera, etcetera.

A series is simply a sequence that we add up as we go...

2 + 4 + 6 + 8 + ... is a series.

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Terms

Each number in a sequence is called a term. The first number is called the first term and so on.

Using algebra, we could call the first term "a", the second "b", the third "c" and so on but there are a couple of problems with this.

Problem 1... it is not easy to find what we'd call the 18th number - unless you're very quick at working out your alphabet. Problem 2... more serious is the problem that we run out of letters at "z".

Instead, we call the first term u₁, the second term u₂ , the 850th term u₈₅₀ and so on. But just to confuse you we also call the first term a when we feel like it.

n-th Term

The first way is to use a function to describe the n-th term. For example, the sequence 3, 5, 7, 9, 11, ... has n-th term, u_n = 2n+1.

This is often the best way to describe a sequence but it is not perfect. Try to describe the n-th term of the Fibonacci Sequence (1, 1, 2, 3, 5, 8, ...)

u_n+1 = f(u_n)

The second way to describe sequences is to use the previous terms to find future ones. In our example above 3, 5, 7, 9, 11, ... each term is always 2 more than the one before. If I told you that we are to start at 3 and keep adding twos then you would get the list correct.

The 5th term equals the fourth term add two. The 86th term equals the 85th term plus two. In general the next term equals the current term plus two. Think of the current term as u_n , then the next term is u_n+1. The formula for this sequence is u_n+1 = u_n + 2.

We must say where we are starting: u₁ = 2.

Now we can write the Fibonacci Sequence...

u_n+2 = u_n+1 + u_n ; where u₁ = 1 and u₂ = 1

What this equation tells us is that if we know the 5th and 6th terms, we can find the 7th by adding (n = 5 in the formula). Of course, there is no way of finding the 1st and 2nd terms so we need to be told the first two numbers .

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Arithmetic Sequences

A special (and very simple) type of sequence is one where the numbers go up (or down) by the same number each time.

eg 3, 5, 7, 9, 11,... goes up in 2s.

eg 8, 15, 22, 29, 36, ... goes up in 7s.

eg 9, 7, 5, 3, 1, -1, ... goes up in -2s.

These are all examples of arithmetic sequences. If I told you to write such a sequence that starts with 9 and goes up in 4s, I bet you could write the first 10 terms.

The first term is 9, the second is 9+4 (13) because we are told that the sequence goes up in 4s. The first ten terms are: 9, 13, 17, 21, 25, 29, 33, 37, 41, 45. EASY!

So, in an arithmetic sequence, all we need is the first term and the number it goes up in and we've got all the information we need. We call the first term a and the number it goes up in d, for difference.

General Term

If the 1st term is a, what's the 2nd? (remember that the list goes up by d each time)

That's right! It's a + d.

So what's the 3rd term? It's d more than the 2nd term: a + d + d which is a + 2d.

The fourth? Yes, it's a + 3d.

Here's a table...

Term	1st	2nd	3rd	4th	5th	6th
Value	a	a + d	a + 2d	a + 3d	a + 4d	a + 5d

Notice that to get to the 6th term, we only need to add d 5 times.

In general the n-th term is... u_n = a + (n -1)d

eg. In the arithmetic sequence with first term 9 and common difference 3, find the 21st term

Answer... u₂₁ = u₁ + 20d

= 9 + 20x3 = 69.

Sum of Terms

Right folks, time for a clip...

Sigma Notation

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