Instruction
1
The average arithmetic mean of a set of numbers is defined as their sum divided by their number. The sum of all numbers of the set divided by the number of numbers in this set.The most simple case is to find the arithmetic mean of two numbers x1 and x2. Then their arithmetic mean X = (x1+x2)/2. For example, X = (6+2)/2 = 4 - the arithmetic mean of the numbers 6 and 2.
2
The General formula for finding the arithmetic mean of the n numbers would look like this: X = (x1+x2+...+xn)/n. It can also be written as: X = (1/n)?xi, where the summation is for index i from i = 1 to i = n.For example, the arithmetic mean of three numbers X = (x1+x2+x3)/3, five numbers is (x1+x2+x3+x4+x5)/5.
3
Interest is the situation when the set of numbers represents the members of an arithmetic progression. As you know, members of the arithmetic progression is a1+(n-1)d, where d is the step of progression and n is the number of the member of progression.Let a1, a1+d, a1+2d,..., a1+(n-1)d - members of an arithmetic progression. Their arithmetic mean 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)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 mean of members of an arithmetic progression is equal to the arithmetic mean of its first and last members.
4
It is also true the property that each member of an arithmetic progression is equal to the average between the previous and the next member of the progression: an = (a(n-1)+a(n+1))/2 where a(n-1), an, a(n+1) successive members of the sequence.