Sequences

A sequence is a list of numbers that are in order and follow certain pattern or rule.

When the sequence goes on forever it is called an infinite sequence, otherwise it is a finite sequence.

Each element in a sequence is called a term.

Example of a finite sequence:
$\{1,4,7,10\}$

A sequence is like a set, except:

• the terms are in order (with sets the order does not matter)
• the same value can appear many times (only once in sets)

A sequence can be defined by an explicit formula or by a recursive one.

Explicit formula

The explicit formula of the above sequence would be:

$\{a_k{\}}_{k=1}^4=1+3(k-1)$

Recursive formula

The recursive formula would be:

$\{a_k{\}}_{k=1}^4$ with $\{\begin{array}{l}{a}_1=2\\ a_k=a_{k-1}+3\end{array}$

Types of sequences

Arithmetic sequence

In an arithmetic sequence the difference between one term and the next is a constant.

In other words, we just add some value each time on to infinity (or up to an established number).

In general, we can write an arithmetic sequence like this:

$\{a_1,a_1+d,a_1+2d,a_1+3d,...\}$

Where d is the difference between the terms (called the common difference).

And we can make the rule (explicit formula):

$a_n=a_1+d(n-1)$

(We use "n-1" because d is not used in the 1st term).

Or expressed in a recursive way:

$\{\begin{array}{l}{a}_1=c\\ a_n=a_{n-1}+d\end{array}$

Geometric sequence

In a geometric sequence each term is found by multiplying the previous term by a constant.

In general, we can write a geometric sequence like this:
$\{a_1,a_1\cdot r,a_1\cdot r^2,a_1\cdot r^3,...\}$

Where r is the factor between the terms (called the common ratio).

The explicit formula is defined by:
$a_n=a\cdot r^{n-1}$

(We use $n-1$ because $a\cdot r^0$ is the 1st term).

The recursive formula is defined by:
$\{\begin{array}{l}{a}_1\\ a_i=r\cdot a_{n-1}\end{array}$

An example of this type of sequence could be $\{2,4,8,16,32,64,128,256,...\}$
Defined by $a_n=2^n$

Sequences are functions

Notice that the explicit formulas of sequences work like functions: We input a term number $n$ and the formula outputs the value of that term $a(n)$.

Sequences are in fact defined as functions. However, $n$ cannot be any real number value. The domain of sequences—which is the set of all possible inputs of the function—is the positive integers.

References

Khan academy
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