 # Arithmetic Progression

Sequence: A set of numbers arranged in some definite order and formed according to some rules is called a sequence.

Progression: The sequence that follows a certain pattern is called 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 an we denote the n-th term of an arithmetic progression.

By Sn 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 + an = a2 + an-1 = ... = ak + an-k+1

and

an = ½(an-1 + an+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...

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

an = a1 + (n - 1)d, n = 1, 2, ...

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

S = ½(a1 + l)n

where a1 is the first term and l the last one.

or

S = ½(2a1 + d(n-1))n

(i) If is a1 given, then d = an – an-1 common difference

(ii) If is given, then term is given by

(iii) If a, b, c are in A.P., then 2b = a + c.

(iv) If a sequence has n terms, its term from the end term from the

beginning.

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