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

1, 3, 5, 7, 9, ….
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:

1, 4, 9, 16, 25, ….
This is the sequence of square numbers. And this sequence,

1, -1, 1, -1, 1, -1, …,
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

1, 3, 5, 9
form a finite sequence containing just four numbers. The numbers

1, 4, 9, 16
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 u₁, the second term u₂, and so on. With this same notation, we would write u_n to represent the n-th term in the sequence. So

u₁, u₂, u₃, …, 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 write u₁ for the first term of the sequence, u₂ for the second term, and so on. We write the n-th term as u_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 a₁, a₂, a₃, …, a_n. a₁ is the first term, a₂ is the second term, a₃ is the third and so on.

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

2, 4, 6, 8, 10, …
and the list of odd numbers

1, 3, 5, 7, 9, …
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

1, 1, 2, 3, 5, 8, 13,21, 34, 55, 89, 144, …
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

1, 3, 5, 7, 9, …
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 nth term of a given sequence. Thus, in this example, a₁=1, a₂=3, a₃=5 and so on; the first term is a₁=1, but there is no last term.

The list of positive odd numbers less than 100

1, 3, 5, 7, …, 99
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 nth term a_n. This is also called a formula for the general term. For example,

a_n=2n-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 a₁₀=2×10-1=19.
Sequences can also be used to approximate real numbers. Thus, for example, the terms in the sequence

1, 1.4, 1.41, 1.414, 1.4142, …
give approximations to the real number √2.

Sequences can be either finite or infinite. For example,

2, 4, 6, 8, 10
is a finite sequence with five terms, whereas

2, 4, 6, 8, 10, …
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

1, ½, ⅓, ¼, ⅕, …
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,

2+4+6+8+…+20
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,

1+3+5+7+9+…
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 n².

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.