Problems in arithmetic progression already existed in ancient times. They appeared and demanded a solution, since they had a practical need.
So, in one of the papyri of Ancient Egypt, which has a mathematical content - the papyrus of Rinda (XIX century BC) - contains the following task: divide ten measures of bread into ten people, provided that the difference between each of them is one eighth of the measure. ”
And in the mathematical works of the ancient Greeks there are elegant theorems related to arithmetic progression. So, the Gypsicle of Alexandria (II century BC), which compiled many interesting tasks and added the fourteenth book to the "Beginnings" of Euclid, formulated the idea: "In an arithmetic progression having an even number of members, the sum of the members of the second half is greater than the sum of the members 1- "by the number that is a multiple of the square 1/2 of the number of members."
Take an arbitrary series of natural numbers (greater than zero): 1, 4, 7, ... n-1, n, ..., which is called a numerical sequence.
The sequence an is designated. The numbers of a sequence are called its members and are usually indicated by letters with indices that indicate the serial number of this member (a1, a2, a3 ... read: “a 1st”, “a 2nd”, “a 3rd” and so on )
The sequence can be infinite or finite.
But what is arithmetic progression? It is understood as a sequence of numbers obtained by adding the previous term (n) with the same number d, which is the difference of the progression.
If d <0, then we have a decreasing progression. If d> 0, then such a progression is considered to be increasing.
An arithmetic progression is called finite if only a few of its first members are taken into account. With a very large number of members, this is already an endless progression.
Any arithmetic progression is given by the following formula:
an = kn + b, while b and k are some numbers.
The statement is absolutely true, which is the opposite: if the sequence is given by a similar formula, then this is exactly an arithmetic progression, which has the properties:
- Each member of the progression is the arithmetic mean of the previous member and the next.
- The converse: if, starting from the 2nd, each term is the arithmetic mean of the previous term and the next, i.e. if the condition is satisfied, then this sequence is an arithmetic progression. This equality is at the same time a sign of progression; therefore, it is usually called the characteristic property of progression.
The theorem that reflects this property is true in the same way: a sequence is an arithmetic progression only if this equality is true for any member of the sequence starting from the 2nd.
The characteristic property for any four numbers of arithmetic progression can be expressed by the formula an + am = ak + al if n + m = k + l (m, n, k are the numbers of progression).
In an arithmetic progression, any necessary (Nth) term can be found using the following formula:
an = a1 + d (n – 1).
For example: the first term (a1) in an arithmetic progression is given and equal to three, and the difference (d) is equal to four. You need to find the forty-fifth member of this progression. a45 = 1 + 4 (45-1) = 177
The formula an = ak + d (n - k) allows us to determine the nth term of an arithmetic progression through any of its kth terms, provided that it is known.
The sum of the members of the arithmetic progression (implies the first n members of the final progression) is calculated as follows:
Sn = (a1 + an) n / 2.
If the difference between the arithmetic progression and the first term are known, then another formula is convenient for calculating:
Sn = ((2a1 + d (n – 1)) / 2) * n.
The sum of the arithmetic progression, which contains n members, is calculated as follows:
Sn = (a1 + an) * n / 2.
The choice of formulas for calculations depends on the conditions of the tasks and the source data.
The natural series of any numbers, such as 1,2,3, ..., n, ... is the simplest example of arithmetic progression.
In addition to arithmetic progression, there is also a geometric progression, which has its own properties and characteristics.