When in comprehensive schools they study the properties of ordered sequences of numbers, they necessarily consider the so-called decreasing infinite geometric progression. We will reveal this question in more detail in the article.
What is geometric progression?
Before proceeding to explain the diminishing infinite geometric progression, a definition of this numerical sequence should be given. Geometric progression is a series of numbers in which any subsequent term is uniquely obtained from the previous one by multiplying it by some rational number. This number is called the denominator.
An example of this type of progression is the following series of numbers: 1, 4, 16, 64, ... It can be seen that if we multiply any of these numbers by 4, we get the next member of the series. This means that the denominator of this sequence is found by the formula: r = a n / a n-1 , here a n and a n-1 are the n-th and (n-1) -th terms of the progression.
Based on the definition of this type of progression, you can find the nth member using the following expression: a n = a 1 * r (n-1) , that is, it is enough to know the denominator and the first member of the number series.
For example, we find the 8th number in the geometric progression given above. We have: a 8 = a 1 * r 7 = 1 * 4 7 = 16384.
Another important formula for geometric progression is the expression for finding the sum of its n first members. This formula has the form: S n = a 1 * (r n -1) / (r-1). We apply it to find the sum of 8 numbers from the sequence above. We get: S 8 = 1 * (4 8 -1) / (4-1) = 21845.
What are geometric progressions?
Depending on the sign and the denominator module r, 4 types of geometric progression are distinguished:
- Increasing. If r> 1, then each subsequent term will be larger than the previous one modulo. The infinite sum of such a series tends to infinity (or minus infinity if the 1st term is a negative number). An example of this progression is considered in the previous paragraph.
- Constant. If r = 1, then we have the usual set of identical numbers.
- Variable. If r <0 and | r |> 1, then we get a sequence in which two neighboring terms differ in sign. For example, 1, -3, 9, -27, 81, ... Here r = -3.
- Waning. If | r | <1, then with an increase in the number of the number in the row, its absolute value will decrease. The next row is a striking example of this type of geometric progression: 100, 50, 25, 12.5 ..., where the denominator is r = 0.5.
Further in the article, we consider a decreasing progression as the most interesting and useful numerical series for practice.
Decreasing progression
As mentioned above, the denominator of an infinitely decreasing geometric progression in absolute value must be less than unity, that is, | r | <1. This means that it can be both positive and negative.
Of practical interest is the sum of the terms of the progression of the geometric infinitely decreasing, because it represents a finite number.
To obtain the formula for the case under consideration, we use the expression for the sum that is given in the first paragraph of the article: S n = a 1 * (r n -1) / (r-1). If we consider an infinite series, that is, n> ∞, then r n > 0, since | r | <1. This fact can be verified by taking any number that satisfies the last condition and raising it to a large extent. As a result, the formula for the sum of n terms for n> ∞ for a decreasing progression takes the form: S ∞ = a 1 * / (1-r).
We give an example of using the obtained formula. Let it be necessary to find an infinite sum for a series of 100, 50, 25, 12.5 ... As you can see, the first term of the progressive geometric decreasing infinitely a 1 is 100, and its denominator r = 0.5 (50/100 = 25/50 = 12.5 / 25). Substituting these values in the formula for an infinite sum, we obtain: S ∞ = a 1 * / (1-r) = 100 / (1-0.5) = 200.
Turtle and Achilles (Zeno paradox)
Where can I use the result obtained in paragraph above? For example, when explaining the paradox of the ancient Greek philosopher Zeno. The essence of this paradox is that Achilles (from the ancient Greek language this name is translated as "one who has" light "legs"), being the fastest warrior, cannot catch up with the turtle.
Zeno reasoned as follows: if the turtle is ahead of Achilles, and at the same time they begin to move, then when the warrior reaches the place where the turtle took the start, the latter will already crawl some distance, so Achilles will have to overcome it again (although it is smaller than the original) . Having run a new stretch of the path, the warrior will still be behind the turtle, because she will again crawl some distance. So the way you can argue ad infinitum.
Each of us knows that not only Achilles, but any person moving on foot will overtake the turtle. What is the mistake of the philosopher? He did not take into account that although the sum of the segments is infinite, it leads to a finite number S ∞ . As soon as Achilles overcomes the distance S ∞ , he will immediately overtake the turtle.
It is curious to note that the philosopher himself explained the fact that Achilles in practice nevertheless overtakes the turtle by the fact that movement and time are an illusion and do not exist in reality.