Problems in arithmetic progression existed in ancient times.They appeared and demanded solutions, because they had a practical necessity.
Thus, in one of the papyri of ancient Egypt, having a mathematical content, - the papyrus Rhind (XIX century BC) - contains such a task: Section Ten measures of bread for ten people, provided if the difference between each of them is one-eighth of the measures".
And in mathematical writings of the ancient Greeks found elegant theorems related to an arithmetic progression.For Gipsikl Alexandria (II century BC), amounting to a lot of interesting challenges and added fourteen books to the "beginning" of Euclid, formulated the idea: "In the arithmetic progression having an even number of members, the amount of members of the second half more than the sum of members 1second on a multiple of the square of 1/2 of the members. "
take an arbitrary number of integers (greater than zero), 1, 4, 7, ... n-1, n, ..., which is called the numerical sequence.
refers to a sequence an.Numbers sequence called its members and usually denoted letters with indices, which indicate the sequence number of the member (a1, a2, a3 ... read: «a first», «a second», «a 3-Thiers' and so on).
sequence may be infinite or finite.
And what is arithmetic progression?It is understood as a sequence of numbers is obtained by adding the previous term (n) with the same number of d, which is the difference progression.
If d & lt; 0, we have a decreasing progression.If d & gt; 0, then this is considered an increasing progression.
arithmetic progression is called finite, if we consider only a few of its first members.When a very large number of members it has an infinite progression.
Sets any arithmetic progression following formula:
an = kn + b, b, and thus k - some numbers.
absolutely true statement, which is the reverse: if the sequence is given by a similar formula, it is exactly the arithmetic progression, which has properties:
- Each member of progression - the arithmetic mean of the previous term and then.
- : if, starting from the second, each member - the arithmetic mean of the previous term and then, ieif the condition, this sequence - an arithmetic progression.This equality is both a sign of progress, therefore, commonly referred to as a characteristic property of progression.
Similarly, the theorem is true that reflects this property: the sequence - arithmetic progression only if this equality is true for any of the members of the sequence, starting with the second.
characteristic property of all four numbers arithmetic progression may be expressed by an + am = ak + al, if n + m = k + l (m, n, k - number of progression).
arithmetically any desired (N-th) member can be found by using the following formula:
an = a1 + d (n-1).
For example: the first term of (a1) in an arithmetic progression and is set to three, and the difference (d) equals four.Find necessary to forty-fifth member of this progression.a45 = 1 +4 (45-1) = 177
formula an = ak + d (n - k) to determine the n-th term of the arithmetic progression through any of its k-th member, provided he is known.
sum of terms of an arithmetic progression (meaning the first n terms of the ultimate progression) is calculated as follows:
Sn = (a1 + an) n / 2.
If you know the difference between an arithmetic progression and the first member, is convenient to calculate a different formula:
Sn = ((2a1 + d (n-1)) / 2) * n.
amount arithmetic progression which comprises members n, calculated thus:
Sn = (a1 + an) * n / 2.
Selecting formulas for calculation depends on the objectives and the initial data.
any number of natural numbers, such as 1,2,3, ..., n, ...- simplest example of an arithmetic progression.
In addition there is an arithmetic progression and geometric, which has its own properties and characteristics.
