# Is the following statement true or false: every sequence is either arithmetic or geometric?

Nov 9, 2016

False

#### Explanation:

Here is a sample sequence:
$\textcolor{w h i t e}{\text{XXX}} 3 , 17 , 5 , 8 , 106$

Notice that there is

neither
- a common difference between successive terms (as required for an arithmetic sequence),

nor
- a common ration between successive terms (as required for a geometric sequence).

Nov 9, 2016

False

#### Explanation:

A sequence is a list of items starting with an initial item.

The items may come from any set - they need not be numbers.

The list can terminate, in which case we call it a finite sequence.

If it does not terminate, we call it an infinite sequence.

Most infinite sequences you will encounter are indexed by positive integers, so will have elements:

${a}_{1} , {a}_{2} , {a}_{3} , \ldots$

We can think of this as a mapping from the set of positive integers into a set $A$.

Arithmetic and geometric sequences are very specific kinds of sequences, but they are often encountered so worth knowing well.

Arithmetic sequence

An arithmetic sequence is a sequence of numbers with a common difference. That is, each consecutive pair of terms has the same difference. We can write a formula for the general term of an arithmetic sequence as:

${a}_{n} = a + d \cdot \left(n - 1\right)$

where $a$ is the initial term and $d$ the common difference.

Geometric sequence

A geometric sequence is a sequence of number with a common ratio. That is, each consecutive pair of terms has the same ratio. We can write a formula for the general term of a geometric sequence as:

${a}_{n} = a \cdot {r}^{n - 1}$

where $a$ is the initial term and $r$ the common ratio.

Fibonacci sequence

One famous example of a sequence that is neither arithmetic nor geometric is the Fibonacci sequence, which we can define by:

$\left\{\begin{matrix}{F}_{1} = 1 \\ {F}_{2} = 1 \\ {F}_{n + 2} = {F}_{n} + {F}_{n + 1}\end{matrix}\right.$

It starts:

$1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 , 144 , 233 , 377 , 610 , 987 , \ldots$

This has no common difference or common ratio between terms.