# Infinite Sequences

## Key Questions

• It depends on the type of sequence.

If the sequence is an arithmetic progression with first term ${a}_{1}$, then the terms will be of the form:

${a}_{n} = {a}_{1} + \left(n - 1\right) b$
for some constant b.

If the sequence is a geometric progression with first term ${a}_{1}$, then the terms will be of the form:

${a}_{n} = {a}_{1} \cdot {r}^{n - 1}$
for some constant $r$.

There are also sequences where the next number is defined iteratively in terms of the previous 2 or more terms. An example of this would be the Fibonacci sequence:

$0 , 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 , \ldots$

Each term is the sum of the two previous terms.

The ratio of successive pairs of terms tends towards the golden ratio $\phi = \frac{1}{2} + \frac{\sqrt{5}}{2} \cong 1.618034$

The terms of the Fibonacci sequence are expressible by the formula:

${F}_{n} = \frac{{\phi}^{n} - {\left(- \phi\right)}^{-} n}{\sqrt{5}}$ (starting with ${F}_{0} = 0$, ${F}_{1} = 1$)

In general an infinite sequence is any mapping from $\mathbb{N} \to S$ for any set $S$. It can be defined in any way you like.

Finite sequences are the same, except that they are mappings from a finite subset of $\mathbb{N}$ consisting of those numbers less than some fixed limit, e.g. $\left\{n \in \mathbb{N} : n \le 10\right\}$

• A sequence is an infinite set of points which are ordered and a mapping can be associated with with that set and the set of natural numbers.

The general notion of a sequence is that it is an infinite set with every element associated with a natural number even though all infinite sets may not be sequences.

A sequence may be represented by, $\left\{{x}_{n}\right\}$ where ${x}_{n}$ is the $n$th element related to the a corresponding natural number.

Thus if ${x}_{n} = \frac{1}{n} ^ 2$, the sequence may be given as,

$\left\{1 , \frac{1}{4} , \frac{1}{9} , \frac{1}{16} , \ldots .\right\}$

For ${x}_{n} = {n}^{3}$ we shall have,

$\left\{1 , 8 , 27 , 64 , \ldots .\right\}$

Now for ${x}_{n} = n$ we can have,

$\left\{1 , 2 , 3 , \ldots . .\right\}$

This is indeed the set of naturals.

However, there can be other ordered arrays of numbers which are sometimes referred at as sequences. They don't fit right with the definition I gave.

Let's for example take a Fibonacci sequence.

$1 , 1 , 2 , 3 , 5 , 8 , 13 , \ldots .$

This sequence is made by adding the previous two numbers on the list to form the next one and so on.

There can be arithmetic sequences, like

$2 , 8 , 14 , 20 , \ldots .$ which has first term $2$ and common difference $6$.

I like to call them progressions and reserve the word sequence for the definition I proposed in the first 2 paragraphs.

It depends.

#### Explanation:

There are many types of sequences. Some of the interesting ones can be found at the online encyclopedia of integer sequences at https://oeis.org/

Let's look at some simple types:

Arithmetic Sequences

${a}_{n} = {a}_{0} + \mathrm{dn}$

e.g. $2 , 4 , 6 , 8 , \ldots$

There is a common difference between each pair of terms.

If you find a common difference between each pair of terms, then you can determine ${a}_{0}$ and $d$, then use the general formula for arithmetic sequences.

Geometric Sequences

${a}_{n} = {a}_{0} \cdot {r}^{n}$

e.g. $2 , 4 , 8 , 16 , \ldots$

There is a common ratio between each pair of terms.

If you find a common ratio between pairs of terms, then you have a geometric sequence and you should be able to determine ${a}_{0}$ and $r$ so that you can use the general formula for terms of a geometric sequence.

Iterative Sequences

After the initial term or two, the following terms are defined in terms of the preceding ones.

e.g. Fibonacci

${a}_{0} = 0$
${a}_{1} = 1$
${a}_{n + 2} = {a}_{n} + {a}_{n + 1}$

For this sequence we find: ${a}_{n} = \frac{{\phi}^{n} - {\left(- \phi\right)}^{- n}}{\sqrt{5}}$ where $\phi = \frac{1 + \sqrt{5}}{2}$

There are many ways to make these iterative rules, so there is no universal method to provide an expression for ${a}_{n}$

Polynomial Sequences

If the terms of a sequence are given by a polynomial, then given the first few terms of the sequence you can find the polynomial.

e.g.

$\textcolor{red}{1} , 2 , 4 , 7 , 11 , \ldots$

Form the sequence of differences of these values:

$\textcolor{red}{1} , 2 , 3 , 4 , \ldots$

Form the sequence of differences of these values:

$\textcolor{red}{1} , 1 , 1 , \ldots$

Once you reach a constant sequence like this, pick out the initial terms from each sequence. In this case $1$, $1$ and $1$.

These form the coefficients of a polynomial expression:

a_n = color(red)(1)/(0!) + (color(red)(1)*n)/(1!) + (color(red)(1)*n(n-1))/(2!)

$= {n}^{2} / 2 + \frac{n}{2} + 1$