Sequences

In mathematics, a sequence is an ordered list of objects. Like a set, it contains members (also called elements or terms). The number of ordered elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and a particular term can appear multiple times at different positions in the sequence. 

For example, $(M, A, R, Y)$ is a sequence of letters that differs from $(A, R, M, Y)$, as the ordering matters, and $(1, 1, 2, 3, 5, 8)$, which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers $(2, 4, 6, \cdots )$. Finite sequences are sometimes known as strings or words and infinite sequences as streams.

Sequence

Part of an infinite sequence of real numbers (in blue). This sequence is neither increasing, nor decreasing, nor convergent. It is, however, bounded within the two dashed lines.

Examples and Notation

Finite and Infinite Sequences

A more formal definition of a finite sequence with terms in a set $S$ is a function from $\left \{ 1, 2, \cdots, n \right \}$ to $S$ for some $n > 0$. An infinite sequence in $S$ is a function from $\left \{ 1, 2, \cdots \right \}$ to $S$. For example, the sequence of prime numbers $(2,3,5,7,11, \cdots )$ is the function 

$1\rightarrow 2, 2\rightarrow 3, 3\rightarrow 5, 4\rightarrow 7, 5\rightarrow 11, \cdots$ 

A sequence of a finite length n is also called an $n$-tuple. Finite sequences include the empty sequence $( \quad )$ that has no elements.

Recursive Sequences

Many of the sequences you will encounter in a mathematics course are produced by a formula, where some operation(s) is performed on the previous member of the sequence $a_{n-1}$ to give the next member of the sequence $a_n$. These are called recursive sequences.

Arithmetic Sequences

An arithmetic (or linear) sequence is a sequence of numbers in which each new term is calculated by adding a constant value to the previous term. An example is $(10,13,16,19,22,25)$. In this example, the first term (which we will call $a_1$) is $10$, and the common difference ($d$)—that is, the difference between any two adjacent numbers—is $3$. The recursive definition is therefore 

$\displaystyle{a_n=a_{n-1}+3, a_1=10}$

Another example is $(25,22,19,16,13,10)$. In this example $a_1 = 25$, and $d=-3$. The recursive definition is therefore 

$\displaystyle{a_n=a_{n-1}-3, a_1=25}$

In both of these examples, $n$ (the number of terms) is $6$.

Geometric Sequences

A geometric sequence is a list in which each number is generated by multiplying a constant by the previous number. An example is $(2,6,18,54,162)$. In this example, $a_1=2$, and the common ratio ($r$)—that is, the ratio between any two adjacent numbers—is 3. Therefore the recursive definition is 

$a_n=3a_{n-1}, a_1=2$

Another example is $(162,54,18,6,2)$. In this example $a_1=162$, and $\displaystyle{r=\frac{1}{3}}$. Therefore the recursive formula is 

$\displaystyle{a_n=\frac13\cdot a_{n-1}, a_1=162}$ 

In both examples $n=5$. 

Explicit Definitions

An explicit definition of an arithmetic sequence is one in which the $n$th term is defined without making reference to the previous term. This is more useful, because it means you can find (for instance) the 20th term without finding all of the other terms in between.

To find the explicit definition of an arithmetic sequence, you begin writing out the terms. Assume our sequence is $t_1, t_2, \dots $. The first term is always $t_1$. The second term goes up by $d$, and so it is $t_1+d$. The third term goes up by $d$ again, and so it is $(t_1+d)+d,$  or in other words, $t_1+2d$. So we see that: 

$\displaystyle{ \begin{aligned} t_1 &= t_1 \\ t_2 &= t_1+d \\ t_3 &= t_1+2d \\ t_4 &= t_1+3d \\ &\vdots \end{aligned} }$

and so on. From this you can see the generalization that: 

$t_n = t_1+(n-1)d$

which is the explicit definition we were looking for.

The explicit definition of a geometric sequence is obtained in a similar way. The first term is $t_1$; the second term is $r$ times that, or $t_1r$; the third term is $r$ times that, or $t_1r^2$; and so on. So the general rule is: 

$t_n=t_1 \cdot r^{n-1}$