**Sequences**

What is a sequence? It is a set of numbers which are written in some particular order. For example, take the numbers

Here, we seem to have a rule. We have a sequence of odd numbers. To put this another way, we start with the number 1, which is an odd number, and then each successive number is obtained by adding 2 to give the next odd number.

Here is another sequence:

This is the sequence of square numbers. And this sequence,

is a sequence of numbers alternating between 1 and -1. In each case, the dots written at the end indicate that we must consider the sequence as an infinite sequence, so that it goes on for ever.

On the other hand, we can also have finite sequences. The numbers

form a finite sequence containing just four numbers. The numbers

also form a finite sequence. And so do these, the numbers 1, 2, 3, 4, 5, 6, …,

*n*.

These are the numbers we use for counting, and we have included

*n*of them. Here, the dots indicate that we have not written all the numbers down explicitly. The

*n*after the dots tells us that this is a finite sequence, and that the last number is

*n*.

Here is a sequence that you might recognise: 1, 1, 2, 3, 5, 8, This is an infinite sequence where each term (from the third term onwards) is obtained by adding together the two previous terms. This is called the **Fibonacci sequence**.

We often use an algebraic notation for sequences. We might call the first term in a sequence *u*_{1}, the second term *u*_{2}, and so on. With this same notation, we would write *u*_{n} to represent the *n*-th term in the sequence. So

*u*

_{1},

*u*

_{2},

*u*

_{3}, …,

*u*

_{n}

would represent a finite sequence containing

*n*terms. As another example, we could use this notation to represent the rule for the

**Fibonacci sequence**. We would write

*u*

_{n}=

*u*

_{(n-1)}+

*u*

_{(n-2)}

to say that each term was the sum of the two preceding terms.

Key Point

A sequence is a set of numbers written in a particular order. We sometimes writeu_{1}for the first term of the sequence,u_{2}for the second term, and so on. We write then-th term asu_{n}

A set of numbers arranged in order by some fixed rule is called as **a sequence**.

For example

(i) 2, 4, 6, 8, 10, 12, 14, …

(ii) 1, 3, 5, 7, 9, …

(iii) ½,¼,⅛, 1/16, …

In sequence *a*_{1}, *a*_{2}, *a*_{3}, …, *a*_{n}. *a*_{1} is the first term, *a*_{2} is the second term, *a*_{3} is the third and so on.

We encounter sequences at the very beginning of our mathematical experience. The list of even numbers

and the list of odd numbers

are examples. We can ‘predict’ what the 20th term of each sequence will be just by using common sense.

Another sequence of great historical interest is the **Fibonacci sequence**

in which each term is the sum of the two preceding terms; for example, 55=21+34. In this case it is somewhat more difficult to predict the 20th term, without listing all the previous ones.

The list of positive odd numbers

is an example of a typical

**infinite sequence**. The dots indicate that the sequence continues forever, with no last term. We will use the symbol

*a*

_{n}to denote the

*n*th term of a given sequence. Thus, in this example,

*a*

_{1}=1,

*a*

_{2}=3,

*a*

_{3}=5 and so on; the first term is

*a*

_{1}=1, but there is no last term.

The list of positive odd numbers less than 100

is an example of a typical

**finite sequence**. The first term of this sequence is 1 and the last term is 99. This sequence contains 50 terms.

There are several ways to display a sequence:

● write out the first few terms

● give a formula for the general term

● give a recurrence relation.

Writing out the first few terms is not a good method, since you have to ‘believe’ there is some clearly defined pattern, and there may be many such patterns present. For example, if we simply write

1,2,4,…

then the next term might be 8 (powers of two), or possibly 7 (**Lazy Caterer’s sequence**), or perhaps even 23 if there is some more complicated pattern going on. Hence, if the first few terms only are given, some rule should also be given as to how to uniquely determine the next term in the sequence.

A much better way to describe a sequence is to give a formula for the *n*th term *a*_{n}. This is also called a formula for the **general term**. For example,

*a*

_{n}=2

*n*-1

is a formula for the general term in the sequence of odd numbers 1,3,5, …. From the formula, we can, for example, write down the 10th term, since

*a*

_{10}=2×10-1=19.

Sequences can also be used to approximate real numbers. Thus, for example, the terms in the sequence

give approximations to the real number √2.

Sequences can be either **finite** or **infinite**. For example,

is a finite sequence with five terms, whereas

continues without bound and is an infinite sequence. We usually use to denote that the sequence continues without bound.

For a given infinite sequence, we can ask the questions:

● Can we find a formula for the general term of the sequence?

● Does the sequence have a limit, that is, do the numbers in the sequence get as close as we like to some number?

For example, we can see intuitively that the terms in the infinite sequence

whose general term is 1/

*n*, are approaching 0 as

*n*becomes very large.

A **finite series** arises when we add the terms of a finite sequence. For example,

is the series formed from the sequence 2, 4, 6, 8, …, 20.

An **infinite series** is the ‘formal sum’ of the terms of an infinite sequence. For example,

is the series formed from the sequence of odd numbers. We can spot an interesting pattern in this series. The sum of the first two terms is 4, the sum of the first three terms is 9, and the sum of the first four terms is 16. So we guess that, in general, the sum of the first

*n*terms is

*n*

^{2}.

For a given infinite series, we can ask the questions:

● Can we find a formula for the sum of the first *n* terms of the series?

● Does the series have a limit, that is, if we add the first *n* terms of the series, does this sum get as close as we like to some number as *n* becomes larger?

If it exists, this limit is often referred to as the **limiting sum** of the infinite series. In this module, we examine limiting sums for one special but commonly occurring type of series, known as a geometric series.

Sequences and series are very important in mathematics and also have many useful applications, in areas such as finance, physics and statistics.