Geometric progression and its properties

Geometric progression is important in mathematics as a science, and in applied meaning, since it has an extremely wide scope, even in higher mathematics, say, in the theory of series. The first information about the progressions reached us from Ancient Egypt, in particular, in the form of a known task from the papyrus of Rhind about seven persons having seven cats. Variations of this task were repeated many times at different times in other nations. Even the great Leonardo of Pisa, better known as the Fibonacci (XIII century), turned to her in his "Book of the Abacus".

So, the geometric progression has an ancient history. It is a numerical sequence with a nonzero first term, and each subsequent one, starting with the second one, is determined by the recurrence formula by multiplying the previous one by a constant nonzero number, which is called the denominator of the progression (it is usually denoted by the letter q).
Obviously, it can be found by dividing each successive member of the sequence by the previous one, that is, z 2: z 1 = ... = zn: z n-1 = .... Therefore, to determine the progression (zn) itself, it is sufficient that the value of its first term y 1 and the denominator q be known.

For example, suppose z 1 = 7, q = - 4 (q <0), then the following geometric progression is obtained: 7, - 28, 112, - 448, .... As we see, the sequence obtained is not monotonic.

Recall that an arbitrary sequence is monotonic (increasing / decreasing), when each of its subsequent terms is greater than / less than the previous one. For example, sequences 2, 5, 9, ... and -10, -100, -1000, ... are monotonous, the second of which is a decreasing geometric progression.

In the case when q = 1, in the progression all terms are equal and it is called constant.

In order for a sequence to be a progression of this type, it must satisfy the following necessary and sufficient condition, namely: starting from the second, each of its members must be the geometric mean of the neighboring terms.

This property allows us to find an arbitrary term of the progression with known two nearby ones.

The n-th term of the geometric progression is easily found from the formula: zn = z 1 * q ^ (n-1), knowing the first term z 1 and the denominator q.

Since the numerical sequence has a sum, a few simple calculations will give us a formula that allows us to calculate the sum of the first terms of the progression, namely:

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

Replacing zn with its expression z 1 * q ^ (n-1) in the formula, we get the second formula for the sum of this progression: S n = - z1 * (q ^ n - 1) / (1 - q).

The following interesting fact is worthy of attention: a clay tablet found during excavations of Ancient Babylon, which dates back to the 6th century. BC, remarkably contains the sum of 1 + 2 + 22 + ... + 29, equal to 2 in the tenth degree minus 1. The solution of this phenomenon has not yet been found.

We note one more property of the geometric progression - the constant product of its terms, located at an equal distance from the ends of the sequence.

Of particular importance from a scientific point of view is the notion of an infinite geometric progression and the calculation of its sum. If we assume that (yn) is a geometric progression having a denominator q satisfying the condition | q | <1, then its sum will be the limit to which the sum of its first terms known to us tends, given an unbounded increase of n, that is, with its Approaching infinity.

Find this sum in the end with the help of the formula:

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

And, as practice showed, behind the apparent simplicity of this progression is hidden a huge applied potential. For example, if we construct a sequence of squares by the following algorithm, connecting the midpoints of the sides of the previous one, then their areas form an infinite geometric progression having the denominator 1/2. The same progression is formed by the areas of triangles obtained at each stage of construction, and its sum is equal to the area of the original square.