PROGRESSIVE ELEMENTS DETERMINANT.

It has been to construct a linear system from the equations, such that the coefficients of its variable constitute a finite progression when tabulated start from its first element until its last element . The determinant, which includes those terms, called the progressive elements determinant. The finite progression that is mentioned above may be arithmetic progression, or geometric progression, and may be increasing or decreasing. In this paper the values of these determinants were deduced, and formulated ways to find their values.

It has been to construct a linear system from the equations, such that the coefficients of its variable constitute a finite progression when tabulated start from its first element until its last element . The determinant, which includes those terms, called the progressive elements determinant. The finite progression that is mentioned above may be arithmetic progression, or geometric progression, and may be increasing or decreasing. In this paper the values of these determinants were deduced, and formulated ways to find their values.

Introduction:-
The linear system can be created, such that the variables coefficients constitute a square matrix [1], [2], and if the tabulated start from the first element until the last element respectively, may be constituted Arithmetic Progression (A.P.), or Geometric Progression (G.P.). In general, the Arithmetic progression is a sequence of numbers such that the deference, d, between any term and its predecessor is the same, and the geometric progression is a sequence of numbers such that the ratio, r, of any term to its predecessor is the same [3], [4]. In any way, the (A.P.) and (G.P.) are finite progressions.
Consider the following linear system: . It can be solved by Cramer's rule.
Since is (A.P.) or (G.P.), and has common ratio, or common difference. The order (n) of the determinant is split the terms of the progression into n rows and n column, each one of them containing n terms. The elements of the determinant mentioned are read a row after a row. It is valuable to understand that the progression (A.P.) or (G.P.) in determinant are classified into rows, this act allows for the generalizations about the paper subject.

Definitions And Basics Definition (2.1):
The Progression Elements Determinant,(PED), is the determinant whose elements are starting from the first element until the last element constituted Arithmetic progression or Geometric Progression, increasing or decreasing.

Formulas of the Progression Elements Determinant:
The subject of this paper confounds two topics: determinants and progressions, so we need to formation formulas and symbols on different PEDs. Consider that is order of PED, N is the terms number of the progression (elements number of PED), such that , and are finite numbers; so one can adopt the following formulas: i) The PED with Arithmetic Progression denoted by , so its general form given by:

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Such that the first element of the PDF, i.e. the first term of (A.P.), and is the Common Difference.
ii) The PED with Geometric Progression denoted by , so its general form given by:

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Such that the first element of the PDF, i.e. the first term of (G.P.), and is the Common Ratio.
iii) The value of any PED given by .

iv) And for any PED:
The Row (Column) Progressions Let [PED] be a square matrix, its transpose, then: Certainly, that obtaining of the will change the fashion progression terms by interchange rows and the corresponding columns. This change causes the progression to be cut off, Thus progressions are created in each row or in each column, which are not interrelated, but the common deference or common ratio , will keep its value. So we can indicate to this kind of PED by: That is tell us the progressions are separated .i.e. they not consist from its first element until its last element . The formula (4) is a generalization of the following two formulas: with or with for every row in PED. And , with or with for every column in PED.
The may be or not, so the regularity of the all determinant elements without cutting the gradation is not necessary condition to construct a PED. To distinguish the two types of progressions that are mentioned in formulas (6) and (7), we called them: row progressions, and column progressions, respectively. One can refer to them in more detail as in the following formulas:

Definition:
The Row Progressions are collection of progressions elements determinant, each one of them exists in a row of the determinant with common deference (ratio). However, the deference (ratio) between last term of progression and first term of the next progression is not same.
The terms of the Row Progressions are given by: . So, the elements of given by: . n is order of the determinant. In addition, the elements of denoted by: ……….. (10) 670

Definition:
The column Progressions are collection of progressions elements determinant, each one of them exists in a column of the determinant with common deference (ratio). However, the deference (ratio) between last term of progression and first term of the next progression progression is not same. The terms of the Row Progressions are given by: .
So, the elements of denoted by: …… (11) Since is number of column, n is order of the determinant. In addition, the elements of are denoted by: …… (12) Since i and j are as in the formulas (9), (10), and(11).

Evaluation Of Progressive Elements Determinant
Because of the different types of PED, the rules are needed to evalute them, and covering these types.

Rule
If the order of PED is then: i) | | , is the common difference.