Numerical sequences: 1st grade math course to download in PDF.

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1 I. Numerical sequences
- 1.1 1. definition and vocabulary
- 1.2 2. Notations and vocabulary
2 II. Various ways to define a sequence
- 2.1 1. Suites defined by a functional equality
- 2.2 2. sequence defined by a recurrence formula
3 III. Arithmetic and geometric sequences:

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 $\mathbb{N}$ to $\mathbb{R}$ , $u:n,\mapsto ,u(n)$ .

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: u_n .

With this notation the image of 0 is u_0 .
We call , the first term of the sequence (u_n) .

Similarly, is the second term of the sequence s.

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

We also write to indicate that it is the sequence whose rank n term is u_n where n $\in$ $\mathbb{N}$ .

Note:

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

$u_n=\frac{1}{n}$ does not exist for n = 0.

The suite starts at row 1.

We will write n $\in$ $\mathbb{N}$ .

$t_n=\frac{1}{n(n-1)}$ 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 $\mathbb{N}$ 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 $\mathbb{N}$ , so it is the restriction to $\mathbb{N}$ of a function defined on $\mathbb{R}$ or a subset of $\mathbb{R}$ containing $\mathbb{N}$ .
For example, the sequence u_n=n^2 (n $\in$ $\mathbb{N}$ ), is the restriction to $\mathbb{N}$ of the function f defined on $\mathbb{R}$ 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 $u_{n+1}$ according to the preceding term by a formula called formula of recurrence.

Definition:

More precisely, the sequence (u_n) will be defined by recurrence by:
– His first term u_0 .
– An equality connecting any two consecutive terms of the sequence $u_{n+1}=f(u_n)$ .

Example:

For example, the sequence defined by its first term and the recurrence formula verified for any integer n: $u_{n+1}=\sqrt{u_n}$ .

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 term_Un = 2 is the geometric sequence of first term 2 and reason 1.
The sequence of general term_Un = (-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 term_Un =^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 (u_n) is an arithmetic sequence of first term u_0 and reason r, we have
For any natural number n, we have:

$\sum_{k=o0}^{n}u_k=u_0+u_1+...+u_n=(n+1)\times ,\frac{u_0+u_n}{2}$

3.sum of the first n integers

Ownership:

$\sum_{k=0}^{n}k=1+2+3+4+...+n=\frac{n(n+1)}{2}$

4.sum of the terms of a geometric sequence

Ownership:

For any natural number n, and for any real $q\neq,1$ , we have

$\sum_{k=0}^{n}q^k=1+q+q^2+....+q^n=\frac{q^{n+1}-1}{q-1}$

Mnemonic:

$\frac{suivant-premier}{raison-1}$ .

Ownership:

If (u_n) is a geometric sequence with first term u_0 and reason $q\neq,1$ , we have:

$\sum_{k=o0}^{n}u_k=u_0+u_1+...+u_n=,\frac{u_{n+1}-u_0}{q-1}$ .

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Numerical sequences: 1st grade math course to download in PDF.

I. Numerical sequences

1. definition and vocabulary

2. Notations and vocabulary

II. Various ways to define a sequence

1. Suites defined by a functional equality

2. sequence defined by a recurrence formula

III. Arithmetic and geometric sequences:

1. definitions and formulas

2.sum of the terms of an arithmetic sequence

3.sum of the first n integers

4.sum of the terms of a geometric sequence

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