Geometric progression and its properties

Geometric progression is important in mathematics as a science, and applied significance, since it has a very broad scope, even in higher mathematics, say, the theory of series.The first information on the progress came to us from ancient Egypt, particularly in the form of a well-known problem of the Rhind papyrus seven persons with seven cats.Variations of this problem repeated many times at different times from other nations.Even the great Leonardo of Pisa, better known as Fibonacci (XIII c.), Spoke to her in his "Book of the abacus."

So, geometric progression has an ancient history.It is a numeric sequence with nonzero first term and each subsequent starting from the second, is determined by multiplying the previous recurrence formula for permanent, non-zero number, which is called the denominator progression (it is usually denoted by using the letter q).
Obviously, it can be found by dividing each subsequent term of the sequence to the previous, i.e. two z: z 1 = ... = zn: z n-1 = ....Consequently, the task of the progression (zn) is enough to know the value of it was the first member of y 1 and the denominator q.

example, let z 1 = 7, q = - 4 (q & lt; 0), then we have the following geometrical progression 7 - 28, 112 - 448, ....As you can see, the resulting sequence is not monotonic.

Recall that an arbitrary sequence of monotonous (increasing / decreasing) when each of its future members of more / less than the previous one.For example, the sequence 2, 5, 9, ... and -10, -100, -1000, ... - monotonous, the second of them - is decreasing exponentially.

In the case where q = 1, all members in the progression are obtained equal and it is called constant.

To sequence was progression of this type, it must satisfy the following necessary and sufficient condition, namely: starting from the second, each of its members should be the geometric mean of neighboring Member States.

This property allows under certain two adjacent finding arbitrary term progression.

n-th term of a geometric progression is easy to find the formula: zn = z 1 * q ^ (n-1), knowing the first term z 1 and the denominator q.

Since the numerical sequence is worth, a few simple calculations give us a formula to calculate the sum of the first terms of progression, namely:

S n = - (zn * q - z 1) / (1 - q).

Replacing in the formula value zn its expression z = 1 * q ^ (n-1) to give a second amount of the progression of the formula: S n = - z1 * (q ^ n - 1) / (1 - q).

worthy of attention the following interesting fact: the clay tablet found in excavations of ancient Babylon, which refers to the VI.BC remarkably contains the sum of 1 + 2 + 22 ... + 29 equal to 2 in the tenth power minus 1. The explanation of this phenomenon is not found.

We note one of the properties of geometric progression - a constant work of its members, spaced at equal distance from the ends of the sequence.

particularly important from a scientific point of view, such a thing as an infinite geometric progression and calculating its amount.Assuming that (yn) - a geometric progression having a denominator q, satisfying the condition | q | & lt;1, it will be called the limit of the sum sought by the already known to us the sum of its first members, with unbounded increase of n, so as it approaches infinity.

find this amount as a result of using the formula:

S n = y 1 / (1- q).

And, as experience has shown, the apparent simplicity of this progression is hidden a huge application potential.For example, if we construct a sequence of squares on the following algorithm, connecting the midpoints of the previous one, then they form a square infinite geometric progression having a denominator 1/2.The same progression form triangles and squares obtained at each stage of construction, and its sum is equal to the area of ​​the original square.