AnnotationsQuit

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 u_{n} is called the term of index n or rank n of the sequence.

Note

The sequence u is also denoted (u_{n})_{n\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
} or simply u_{n}. In addition, u_{n+1} is the index term (n+1), also noted u(n+1).

Example

1/ Let u_{n} be the sequence defined for any natural number n by:

u_{n}=2n+3

- 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.

u_{0}=2\times 0+3=0+3=3.\\ u_{1}=2\times 1+3=2+3=5.\\ u_{2}=2\times 2+3=4+3=7.\\ u_{10}=2\times 10+3=20+3=23.

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

v_{n}=(n+1)^{2}

The same previous steps for this example

v_{0}=(0+1)^{2}=1^{2}=1.\\ v_{1}=(1+1)^{2}=2^{2}=4.\\ v_{2}=(2+1)^{2}=3^{2}=9.\\ v_{10}=(10+1)^{2}=11^{2}=121.

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 u_{n+1} as a function of u_{n} for any natural number n). This relation is called a recurrence relation.

Example

Let (u_{n}) be the sequence defined by u₀=2 and for any natural number n by u_{n+1}=-2u_{n}+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 u_{n+1}

u_{1}=-2u_{0}+3=-2\times 2+3=-4+3=-1.\\ u_{2}=-2u_{1}+3=-2\times (-1)+3=2+3=5.

Sequence defined by an explicit formula

A sequence is defined by an explicit formula when u_{n} is expressed directly as a function of n (u_{n}=f(n)). In this case, each term can be calculated from its index.

Example

Let (u_{n})_{n\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
} be the sequence defined for any natural number n by u_{n}=1+3n. Calculate u₀,u₁,u₂ and u₁₀

The answer: As we said in this case each term can be calculated from its index

u_{0}=1+3\times 0=1+0=1.\\ u_{1}=1+3\times 1=1+3=4.\\ u_{2}=1+3\times 2=1+6=7.\\ u_{10}=1+3\times 10=1+30=31.