Geometric progression is important in mathematics, both in science and in applied value, since it has an extremely wide scope, even in higher mathematics, say, in series theory. The first information about progressions came to us from Ancient Egypt, in particular, in the form of a well-known task from the papyrus Ryande about seven individuals with seven cats. Variations of this task were repeated many times at different times among other peoples. Even the great Leonardo of Pisa, better known as Fibonacci (13th century), turned to her in his Book of the Abacus.
So, geometric progression has an ancient history. It is a numerical sequence with a non-zero first term, and each subsequent one, starting from the second, is determined by the recurrence formula by multiplying the previous one by a constant, non-zero number, which is called the progression denominator (it is usually denoted using the letter q).
Obviously, it can be found by dividing each subsequent member of the sequence by the previous one, that is, z 2: z 1 = ... = zn: z n-1 = .... Therefore, to specify 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 we get the following geometric progression 7, - 28, 112, - 448, .... As you can see, the resulting sequence is not monotonic.
Recall that an arbitrary sequence is monotonic (increasing / decreasing), when each of its subsequent members is more / less than the previous one. For example, sequences 2, 5, 9, ... and -10, -100, -1000, ... are monotone, and the second of them is a decreasing geometric progression.
In the case where 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 its neighboring members.
This property allows one to find an arbitrary member of the progression with known two nearby ones.
The nth term of geometric progression is easily found by 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, several 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 the value of zn in the formula with its expression z 1 * q ^ (n-1), we obtain 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: the clay tablet found during excavations of Ancient Babylon, which dates back to the VI century. BC, miraculously contains the sum of 1 + 2 + 22 + ... + 29, equal to 2 to the tenth power minus 1. The solution to this phenomenon has not yet been found.
We note one more of the properties of geometric progression - a constant product of its members spaced at an equal distance from the ends of the sequence.
Of particular scientific importance is the concept of infinite geometric progression and the calculation of its sum. If we assume that (yn) is a geometric progression with a denominator q satisfying the condition | q | <1, then its sum will be the limit to which the sum of its first terms, which we already know, tends to, with an unlimited increase in n, that is, when approaching infinity.
This amount is finally found using the formula:
S n = y 1 / (1-q).
And, as practice has shown, the apparent simplicity of this progression hides a huge applied potential. For example, if you construct a sequence of squares according to the following algorithm, connecting the midpoints of the sides of the previous one, then their areas form an infinite geometric progression having a denominator of 1/2. The same progression is formed by the area of the triangles obtained at each stage of construction, and its sum is equal to the area of the original square.