Arithmetic progression

An arithmetic progression is a sequence in which each term (except the first term) is obtained by adding a fixed number (positive or negative or zero) to the term immediately preceding it.
Hence, this fixed number becomes the difference of two successive terms.
For this reason it is called as the common difference and is usually denoted by d.

Quantities are said to be in Arithmetic Progression (A.P.) when they increase or decrease by a common difference. The common difference is formed by subtracting any term of the sequence from that which follows it.

Thus, if  t1, t2, t3, . . . . . , tn are the terms in an A.P. and the common difference is 'd', then
t2 = t1 + d d = t2 – t1   
t3 = t2 + d d = t3 – t2   
t4 = t3 + d d = t4 – t3
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tn = tn - 1 + d d = tn – tn - 1
In above, if the first term  t1 = a, then
t2   =   t1 + d   =   a + d   =   a + (2 – 1)d
t3   =   t2 + d   =   [a + d] + d   =   a + 2d   =   a + (3 – 1)d
t4   =   t3 + d   =   [a + 2d] + d  =   a + 3d   =   a + (4 – 1)d
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tn   =   tn – 1 + d   =   a + (n – 1)d
tn is called general term of A.P.
Thus tn   =   a + (n – 1)d
By substituting n = 1, 2, 3, . . . we get   a, a + d, a + 2d, a + 3d, a + 4d, . . . . . . . .
It represents an arithmetic progression where "a" is the first term and "d" is the common difference.
This is called the general form of an A.P.
The common difference (d) is given by tn – tn – 1.