A pattern in which each term except the first is obtained by adding a fixed number (positive or negative) to the previous term is called an Arithmetic Progression (A.P.).
Sequence (Progression): A group of numbers forming a pattern.
Arithmetic Progression (A.P.): A progression in which each term, except the first, is obtained by adding a constant to the previous term. Its terms are denoted by t1, t2, t3, tn, or a1, a2, a3, an.
A sequence is called an arithmetic progression, if there exists a constant d such that a2 - a1 = d, a3 - a2 = d, and so on.
an+1 - an = d
d is called the common difference.
Formation of A.P. or General form of A.P.: If a is the first term and d is the common difference of an A.P., then A.P. is
a, a + d, a + 2d, a + 3d, a + 4d, and so on.
nth term of A.P.: The nth term of the A.P. is given by
tn = a + (n - 1)d
Sometimes nth term is also denoted by an.
Sum of first n terms of an A.P.: The sum of first n terms of an A.P. is
$$ S_n = \frac{n}{2}[2a + (n - 1)d] $$
nth term in terms of Sn: If Sn is the sum of the first n terms of an A.P., then the nth term is given by
tn = Sn - Sn-1
Various terms of an A.P.: 3 consecutive terms are a - d, a, a + d and common difference is d. 4 consecutive terms are a - 3d, a - d, a + d, a + 3d and common difference is 2d.