Is the following statement true or false: every sequence is either arithmetic or geometric?
2 Answers
False
Explanation:
Here is a sample sequence:
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).
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
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
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
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