General information about numerical sequences.
The numerical sequences play an important role, it used to model phenomena in all phields.
Definition of a numerical sequence
Definition :
A sequence u is a function on the set
of natural numbers. The image of the integer natural number n by the sequence u, noted u(n) where
is called the term of index n or rank n of the sequence.
Note :
The sequence u is also denoted or simply
. In addition,
is the index term (n+1), also noted u(n+1).
Example :
1/ Let be the sequence defined for any natural number n by:

- Calculate u₀,u₁,u₂ and u₁₀.
The answer: To calculate the given terms we will remplace the index n by 0,1,2 and 10.

2/ Same question for the sequence () defined for any natural number n by:

The same previous steps for this example

Sequence defined by a recurrence relation
Definition :
A sequence is defined by a recurrence relation when it is defined by giving :
its first term.
a relation that allows you to calculate the next term from each term (Express
as a function of
for any natural number n). This relation is called a recurrence relation.
Example :
Let () be the sequence defined by u₀=2 and for any natural number n by
=-2
+3. Calculate u₁ and u₂.
The answer: To calculate the given term we will remplace the index n by 1,2 in the recurrence relation

Sequence defined by an explicit formula
A sequence is defined by an explicit formula when is expressed directly as a function of n (
=
). In this case, each term can be calculated from its index.
Example :
Let be the sequence defined for any natural number n by
=1+3n. Calculate u₀,u₁,u₂ and u₁₀
The answer: As we said in this case each term can be calculated from its index
