What is a (mathematical) sequence?

A sequence is an ordered list of numbers. It can be understood as a function that maps a real number to every natural number. So for every natural number equation starting from 1 (or 0 sometimes), the function associates a real number. Consider the following example sequence:

    equation

In this example, 1 is the first term in the sequence, 3 is the second term and so on. If we express this sequence as a function equation, we get for the terms above:

(1)   equation

We can calculate the equationth term in this example as equation. So in this case, the function is equation and indeed we if we substitute 1 to 5 for equation, we obtain the same numbers as above.

Example of a sequence

Consider a sequence equation. To find the 10th term (equation) in this sequence, we substitute 10 for equation:

    equation

Sequences in recursive and explicit forms

There are two ways to define a sequence. In the example above equation we used an explicit form. This means that we can directly calculate the equationth term. The recursive form of a sequence describes how the next term can be obtained if the previous term is known. One term of the sequence must be given as well as rule how to obtain the next term from it. The same sequence of our example can be defined recursively as:

(2)   equation

In this case, if we want to find the 10th term, we must start with the first which is given by the definition (2 in this case). Then we can obtain the next term which is equation. We always find the next term by multiplying the previous term by 2 until we find the 10th term which is also 1024 as in our example before.

Example: Fibonacci sequence

One famous sequence is called the Fibonacci sequence and is defined recursively as follows:

(3)   equation

In this case, the recursive rule requires two previously known terms to calculate the next term. This is the reason why two terms are given in the definition. We can apply the rule to find the terms of the sequence which gives us:

    equation

Example: sequence of prime numbers

Some sequences do not have a rule to calculate a term explicitly or recursively. One example is the sequence of all prime numbers. The first few terms of the sequence are:

    equation

To find the other terms of the sequence, we need to analyse every number to see if it is a prime number and part of the sequence or not.