@Reyth

I found this from the Italian site.

Maybe it would help too

"PROGRESSION [/size]. [size=78%][/size][/color][/size]- A [size=78%][/size][/color][/size]lgebra[size=78%][/size][/color][/size] . - In the algebraic analysis we call [size=78%][/size][/color][/size]progression[size=78%][/size][/color][/size] a succession of a finite or infinite number of terms, which is constructed by means of a given law.[size=78%][/size][/color][/size]Arithmetic and geometric progressions are especially used .[size=78%][/size][/color]

[/size]Arithmetic progression . - It is called arithmetic progression or by difference a sequence:[size=78%][/size][/color]

[/size]of numbers such that the difference between a number and the one immediately preceding it is a constant date d , which is called the difference or also the reason of the progression. In other words we have:[size=78%][/size][/color]

[/size]Examples of arithmetic progressions can be easily constructed and are known in mathematics from the earliest antiquity. It is certain that in the Pythagorean school (about 500 BC) it was already observed that the vibration frequencies of a homogeneous string, stretched and fixed in two points, form an arithmetic progression.[size=78%][/size][/color]

[/size]The simplest case of such a progression is in fact offered by the natural series of numbers:[size=78%][/size][/color]

[/size]which constitutes an arithmetic (unlimited) progression of reason 1. A limited progression is instead formed that, for example, from the first 5 integers 1, 2, ..., 5. Likewise is an arithmetic progression the series:[size=78%][/size][/color]

[/size]of the even numbers and in general the series of successive multiples of a given number n . Numerous other examples could be given, since, as we have seen from the cases cited, it is enough to change the reason to obtain, starting from the same number, a new progression.[size=78%][/size][/color]

[/size]Arithmetic progressions enjoy some simple properties.[size=78%][/size][/color]

[/size]First of all: the n - th term of such a progression is obtained from the first, adding to it n - 1 times the reason. In symbols:[size=78%][/size][/color]

[/size]This relationship makes it possible to determine any of the four quantities a [size=78%][/size][/color][/size]1 , a n , n , dwhen three are given.An application of this relation is in the problem of the insertion of k medî arithmetic between two numbers a and b . This problem consists of inserting between two numbers a , b ( a < b) k numbers x 1 , x 2 , ..., x k so that:they form an arithmetic progression.The unknown reason must satisfy the equation b = a + ( k + 1) d , and then we have:It follows:Moreover: For a limited arithmetic progression, the sum of two terms whatever equidistant from the extremes is equal to the sum of the extremes themselves.From here follows the theorem easily: The sum of the first n terms in an arithmetic progression is given by the semisomma of the first and of the n -st term, multiplied by n .Geometric progression . - It is called geometric progression or by quotient a sequence:of numbers such that the quotient between a number and the one immediately preceding it is a constant number q , which is called the reason of the progression.Geometric progressions were known to Greek geometers, and indeed it can almost certainly be said that they were not even unknown to the fact that, in certain circumstances, the sum of an infinite number of quantities in geometric progression can give rise to a finite result. In fact, the well-known paradox of Zeno, consisting in the statement that it is impossible to pass from A to B , without passing through the middle point C of AB and then to the midpoint of the CB segment and so to infinity, leads directly to seek the sum of the infinite numbers (see series ):which constitute a geometric progression of reason = ?. Similarly, Democritus leads the calculation of the pyramid's volume to the sum:In modern mathematics the writing of a fraction in the form of a periodic decimal number is precisely to express it as the sum of infinite numbers constituting a geometric progression. Thus, for example, the numberyou can write:and this is equivalent to saying that it is:The properties of geometric progression are analogous to those of arithmetic progression. So we have the propositions:The term n- th of a geometric progression is equal to the product of the first term for the ( n - 1) -thest power of reason.In a limited geometric progression the product of two terms equidistant from extremes is equal to the product of extremes.Moreover: the product of the first n numbers of a geometric progression is given by the square root of the product of the first and the last term raised to the exponent n .For the sum s n of the first n terms we have instead, calling with a 1 the first term and with q the reason:formula in which if - 1 < q <1 can be made to stretch n to infinity, obtaining as the sum of all the terms of the geometric progression the quantity:The relationships between geometric progression and the arithmetic progression remain heightened by the following proposition (v. Logarithm ): Given a geometric progression whose first term is to 1 and whose reason is q , the sequence of numbers that is obtained by taking the logarithms of terms of it, constitutes an arithmetic progression, whose first term is given by log a 1 and whose reason is log q .In the particular case of geometric progression:taking the logarithms (in base 10) we obtain the arithmetic progression:Other types of progressions . - In the algebraic analysis we often consider, next to the arithmetic and geometric progressions , the harmonic progression , which is defined by the property that the inverse of its terms are in arithmetic progression.A generalization of the arithmetic progression occurs then in the arithmetic progressions of a higher order. It is said that a succession of numbers forms a second-order progression, if the second differences are constant for it, that is, if the differences in the differences between a term and its immediate precedent are constant.A simple example of the progression of the second order occurs in the succession of the squares of the integers:for which the series of raw differences is given by successive odd numbers:while the second differences have the constant value 2.The arithmetic progressions of higher order have some application in the calculation of finite differences and can be, for example, used to construct, by means of summing operations, the tables of the squares, of the cubes, ..., of the n -three powers of whole numbers."[size=78%]