A math course on number sequences in first grade is important for students. This chapter will allow them to progress well.

This lesson involves the following concepts:

• definition of a sequence;
• increasing or decreasing sequence;
• sequence defined by a function;
• recurring suites;
• convergence of a sequence;
• gendarme theorem;
• limit of a sequence.

This lesson on numerical sequences is available as a free PDF download.

## I. Numerical sequences

### 1. definition and vocabulary

Definition:

A numerical sequence is a function from to , .

### 2. Notations and vocabulary

Notations:

The functional writing u(n) is rarely used to designate the image of the natural number n by the function u. The indexed notation is preferred: .

With this notation the image of 0 is .
We call , the first term of the sequence .

Similarly, is the second term of the sequence s.

In general:
is the term of index n or rank n of the sequence .
We also say that is the general term of the sequence .

We also write to indicate that it is the sequence whose rank n term is where n .

Note:

Sometimes the first term of a sequence is not .
For example:

does not exist for n = 0.

The suite starts at row 1.

We will write n.

does not exist for n = 0, nor for n = 1.

The suite starts at row 2.

In all cases of this type, we will specify the subset of where the sequence is defined.

## II. Various ways to define a sequence

### 1. Suites defined by a functional equality

Definition:

A numerical sequence is a function defined on , so it is the restriction to of a function defined on or a subset of containing .
For example, the sequence (n ), is the restriction to of the function f defined on by . The interest of this remark lies in the fact that the properties already studied for the functions of the real variable will be usable for the sequences.

### 2. sequence defined by a recurrence formula

The specificity of the sequences on the functions of the real variable, is that, for any natural number n, its image being “numberable”, we can define the term according to the preceding term by a formula called formula of recurrence.

Definition:

More precisely, the sequence will be defined by recurrence by:
– His first term .
– An equality connecting any two consecutive terms of the sequence .

Example:

For example, the sequence defined by its first term and the recurrence formula verified for any integer n: .

## III. Arithmetic and geometric sequences:

### 1. definitions and formulas

Let n be any natural number:

Examples:

• The sequence of natural numbers is the arithmetic sequence of first term 0 and reason 1.
• The sequence of even natural numbers is the arithmetic sequence of first term 0 and reason 2.
• The sequence of odd natural numbers is the arithmetic sequence of first term 1 and reason 2.
• The sequence defined by the formula:Un = an + b (affine function of n) is the arithmetic sequence of first term U0 = b and reason a.
• The constant sequence of general termUn = 2 is the geometric sequence of first term 2 and reason 1.
• The sequence of general termUn = (-1)n is the geometric sequence of first term U0 = 1 and of reason -1.
• The sequence of powers of a non-zero real number a, of general termUn =an is the geometric sequence of first term U0 = 1 and of reason a.
• The sequence defined by the formula:Un = a bn (exponential function of n) is the geometric sequence of first term U0 = a and reason b (b real non zero).

### 2.sum of the terms of an arithmetic sequence

Ownership:

If is an arithmetic sequence of first term and reason r, we have
For any natural number n, we have:

Ownership:

### 4.sum of the terms of a geometric sequence

Ownership:

For any natural number n, and for any real , we have

Mnemonic:

.

Ownership:

If is a geometric sequence with first term and reason, we have:

.

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