Question

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

Answer 1

False

Explanation:

Here is a sample sequence:
`color(white)("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).

Answer 2

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,...`

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*(n-1)`

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*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:

`{ (F_1 = 1), (F_2 = 1), (F_(n+2) = F_n + F_(n+1)) :}`

It starts:

`1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987,...`

This has no common difference or common ratio between terms.

Further reading

The online encyclopedia of integer sequences catalogs many interesting integer sequences. It can be found at http://oeis.org