Arithmetic average - the important concept used in many sections of mathematics and its applications: to statistics, probability theory, economy and. etc. The arithmetic average can be defined as the general concept of average size.
Instruction
1. The arithmetic average of set of numbers is defined as their sum divided into their quantity. That is the sum of all numbers of set is divided into quantity of numbers in this set. The simplest case - to find an arithmetic average of two numbers of x1 and x2. Then their arithmetic average of X = (x1+x2)/2. For example, X = (6+2)/2 = 4 - an arithmetic average of numbers 6 and 2.
2. The general formula for finding of an arithmetic average of n of numbers will look so: X = (x1+x2+...+xn)/n. It can also be written down in a look: X = (1/n)? xi where summation is conducted by index i from i = 1 to i = by n. For example, an arithmetic average of three numbers X = (x1+x2+x3)/3, five numbers - (x1+x2+x3+x4+x5)/5.
3. The situation when a set of numbers represents members of an arithmetic progression is of interest. It is known that members of an arithmetic progression are equal to a1+ (n-1) d where d is a progression step, and n is number of the member of a progression. Let a1, a1+d, a1+2d..., a1+(n-1)d - members of an arithmetic progression. Their arithmetic average is equal to S = (a1+a1+d+a1+2d+...+a1+(n-1)d)/n = (na1+d+2d+... + (n-1)d) / n = a1+ (d+2d+... + (n-2) of d+ (n-1) d) / n = a1+(d+2d+...+dn-d+dn-2d)/n = a1+ (n*d * (n-1)/2) / n = a1+dn/2 = (2a1+d (n-1))/2 = (a1+an)/2. Thus the arithmetic average of members of an arithmetic progression is equal to an arithmetic average of his first and last members.
4. The property is also fair that each member of an arithmetic progression is equal to an arithmetic average of the previous and subsequent member of a progression: an = (a(n-1)+a(n+1))/2, where a(n-1), an, a(n+1) - the members of the sequence going one after another.