Arithmetic progression

Problems on the arithmetic progression existed already in ancient times. They appeared and demanded solutions, because they had a practical need.

Thus, in one of the papyri of ancient Egypt, which has a mathematical content, the Rhindus papyrus (XIX century BC) - contains such a task: stripped ten measures of bread for ten people, provided that the difference between each of them is one eighth measure. "

And in mathematical works of the ancient Greeks there are elegant theorems related to the arithmetic progression. Thus, the Gipsicle of Alexandria (II century BC), which compiled many interesting problems and added the fourteenth book to Euclid's Principles, formulated the idea: "In an arithmetic progression having an even number of terms, the sum of the members of the second half is greater than the sum of the terms of the 1- By a number that is a multiple of the square of 1/2 of the number of terms. "

We take an arbitrary series of positive integers (greater than zero): 1, 4, 7, ... n-1, n, ..., which is called a numerical sequence.

The sequence an. The numbers of a sequence are called its members and are usually denoted by letters with indices that indicate the serial number of this member (a1, a2, a3 ... read: "a 1st", "a 2nd", "a 3-y" and so on ).

The sequence can be infinite or finite.

And what is an arithmetic progression? It is understood as the 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 said to be finite if only a few of its first terms are taken into account. With a very large number of members, this is an infinite progression.

Any arithmetic progression is given by the following formula:

An = kn + b, with b and k being some numbers.

The statement that is the reverse is absolutely true: if a sequence is given by a similar formula, then this is exactly an arithmetic progression that has the properties:

1. Each member of the progression is the arithmetic mean of the previous term and the subsequent one.
2. Conversely, if, starting with the 2nd, each term is the arithmetic mean of the previous term and the next one, i.e. If the condition is satisfied, then this sequence is an arithmetic progression. This equality is also a sign of progression, therefore, as a rule, it is called the characteristic property of progression.
   Similarly, a theorem that reflects this property is true: a sequence is an arithmetic progression only if this equality is true for any of the terms of the sequence, starting with the 2nd.

The characteristic property for any four numbers of an arithmetic progression can be expressed by the formula an + am = ak + al if n + m = k + l (m, n, k are the progression numbers).

In an arithmetic progression, any necessary (N-th) term can be found by applying the following formula:

An = a1 + d (n-1).

For example: the first term (a1) in the arithmetic progression is given and equal to three, and the difference (d) is equal to four. 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 k-th terms, provided it is known.

The sum of the terms of the arithmetic progression (we mean the first n terms of the finite progression) is calculated as follows:

Sn = (a1 + an) n / 2.

If the difference between the arithmetic progression and the first term is known, then another formula is convenient for computing:

Sn = ((2a1 + d (n-1)) / 2) * n.

The sum of the arithmetic progression, which contains n terms, is calculated thus:

Sn = (a1 + an) * n / 2.

The choice of formulas for calculations depends on the conditions of the tasks and the initial data.

The natural series of any numbers, such as 1,2,3, ..., n, ... is the simplest example of an arithmetic progression.

In addition to the arithmetic progression, there is also a geometric progression, which has its own properties and characteristics.