What is a sequence?

Question

What is a sequence?

Klecanlh · Accepted Answer

A sequence is an infinite set of points which are ordeblack 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,

{x_{n}}

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,

{1, \frac{1}{4}, \frac{1}{9}, \frac{1}{16}, ... .}

For

x_{n} = n^{3}

we shall have,
{1,8,27,64,....}
Now for

x_{n} = n

we can have,
{1,2,3,.....}
This is indeed the set of naturals.
However, there can be other ordeblack arrays of numbers which are sometimes referblack 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,....
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,.... 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.

Linda Peters · Accepted Answer

A sequence is simply an ordeblack list like:

{1, 2, 3, 4, ...}

and

{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, ...}

What is a sequence?

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Answer & Explanation

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