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Sequence

A sequence is a list of numbers in a certain order. Each number in a sequence is called a term . Each term in a sequence has a position (first, second, third and so on).

For example, consider the sequence $\{5,15,25,35,\dots \}$

In the sequence, each number is called a term. The number $5$ has first position, $15$ has second position, $25$ has third position and so on.

The $n^{\text{th}}$ term of a sequence is sometimes written $a_n$ .

Often, you can find an algebraic expression to represent the relationship between any term in a sequence and its position in the sequence.

In the above sequence, the $n^{\text{th}}$ term $a_n$ can be calculated using the equation $a_n=10n-5$ .

Finite and Infinite Sequences

A sequence is finite if it has a limited number of terms and infinite if it does not.

Finite sequence: $\left\{4,8,12,16,\dots ,64\right\}$

The first of the sequence is $4$ and the last term is $64$ . Since the sequence has a last term, it is a finite sequence.

Infinite sequence: $\left\{4,8,12,16,20,24,\dots \right\}$

The first term of the sequence is $4$ . The "..." at the end indicates that the sequence goes on forever; it does not have a last term. It is an infinite sequence.

Increasing and Decreasing Sequences

An increasing sequence is one in which every term is greater than the previous term. That is, $a_{n+1}>a_n$ .

The following two sequences are both increasing.

$\left\{5,7,9,11,13,15,\dots \right\}$

$\left\{1,1.5,1.75,1.825,1.9375,\dots \right\}$

A decreasing sequence is one in which every term is greater than the previous term. That is, $a_{n+1}<a_n$ .

The following two sequences are both decreasing.

$\left\{100,50,0,-50,-100,-150,-200,\dots \right\}$

$\left\{1,0.5,0.25,0.125,0.0625,\dots \right\}$

It is possible for a sequence to be neither increasing nor decreasing:

$\left\{0,1,-2,3,-4,5,-6,7,\dots \right\}$

Arithmetic and Geometric Sequences

An arithmetic sequence is a sequence in which the difference between any two consecutive terms is the same.

Example: $10,20,30,40,50,\dots$

Here, the common difference between any two consecutive terms is $10$ .

A geometric sequence is a sequence in which the common ratio between any two consecutive terms is the same.

Example: $2,8,32,128,512,\dots$

Here, the common ratio between any two consecutive terms is $4$ .