Arithmetic Progression

An arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant.
For example, the sequence 1, 2, 3, 4, ... is an arithmetic progression with common difference 1.
Second example: the sequence 3, 5, 7, 9, 11,... is an arithmetic progression
with common difference 2.
Third example: the sequence 20, 10, 0, -10, -20, -30, ... is an arithmetic progression
with common difference -10.

Notation

We denote by d the common difference.
By a_n we denote the n-th term of an arithmetic progression.
By S_n we denote the sum of the first n elements of an arithmetic series.
Arithmetic series means the sum of the elements of an arithmetic progression.

Properties

a₁ + a_n = a₂ + a_n-1 = ... = a_k + a_n-k+1

and

a_n = ½(a_n-1 + a_n+1)

Sample: let 1, 11, 21, 31, 41, 51... be an arithmetic progression.
51 + 1 = 41 + 11 = 31 + 21
and
11 = (21 + 1)/2
21 = (31 + 11)/2...

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If the initial term of an arithmetic progression is a₁ and the common difference of successive members is d, then the n-th term of the sequence is given by

a_n = a₁ + (n - 1)d, n = 1, 2, ...

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The sum S of the first n numbers of an arithmetic progression is given by the formula:

S = ½(a₁ + a_n)n

where a₁ is the first term and a_n the last one. or

S = ½(2a₁ + d(n-1))n