# Sequence

In math, a sequence is a list of objects, typically numbers, in which order matters, repetition is allowed, and the same elements can appear multiple times at different positions in the sequence. They follow what can be referred to as a rule, which enables you to determine what the next number in the sequence is.

For example, the following is a simple sequence comprised of natural numbers that starts from 1 and increases by 1:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10 …

Each number in this sequence is commonly referred to as an element, term, or member. The “…” at the end signifies that the sequence continues infinitely. A finite sequence may be written as follows:

{2, 4, 6, 8}

The sequence above is a sequence of the first 4 even numbers.

Sequences are similar to sets, except that order is important in a sequence. Using the example above, for a sequence, it is important that the numbers are written as:

{2, 4, 6, 8}

For a set however, the numbers could be written the exact same way as above, or as

{2, 6, 8, 4}

{6, 2, 4, 8}

...

or any other combination of those four numbers.

The above sequences are simpler sequences, but there are sequences that are defined by significantly more complex rules. In such cases, and to be able to identify the n^{th} term in a sequence, we need to use certain notations and formulas.

## Sequence notation

The variable n is used to refer to terms in a sequence. For the sequence below,

{2, 4, 6, 8 ... x_{n}}

The terms can be referred to as x_{n} where n refers to the term's position in the sequence. For the above sequence,

x_{1} = 2

x_{2} = 4

x_{3} = 6

x_{4} = 8

...

x_{n} = 2n

For the sequence above, we can see that the pattern is all the even numbers. Thus, the equation for this sequence can be written as:

x_{n} = 2n

This allows us to determine any term in the sequence, where x_{n} is the term, and n is the term number, or position of the term in the sequence.

There are many more complex sequences, and it is possible for a given sequence to be able to be defined using different rules or equations, but these are the basics of sequences.