Arithmetic Progression (AP)

An arithmetic progression is a sequence in which terms increase or decrease by a constant number called the common difference.
(i) The sequence 2, 6, 10, 14, 18, 22… is an arithmetic progression whose first term is 2 and common difference 4.
(ii) The sequence 2,5/2,3,7/2,4…is an arithmetic progression whose first term is 2 and common difference ½.
An arithmetic progression is represented by a, (a + d), (a + 2d), (a + 3d) ....... + (n – 1)d
Here, a = first term
d = common difference
n = number of terms in the progression

• The general term of an arithmetic progression is given by T_n = a + (n – 1) d.
• The sum of n terms of an arithmetic progression is given by S, = [2a + (n – 1) d] or S_n = 2 [a + l]
  where l is the last term of arithmetic progression.
• If three numbers are in arithmetic progression, the middle number is called the arithmetic mean of the other two terms.
• If a, b, c are in arithmetic progression, then b = (a+b)/2 where b is the arithmetic mean.
• Similarly, if ‘n’ terms a_l, a₂, a₃… an are in AP, then the arithmetic mean of these ‘n’ terms is given by
  AM = (a₁+a₂+a₃+.....+a_n)
• If the same quantity is added or multiplied to each term of an AP, then the resulting series is also an AP.
• If three terms are in AP, then they can be taken as (a – d), a, (a + d).
• If four terms are in AP, then they can be taken as (a – 3d), (a – d), (a + d), (a + 3d).
• If five terms are in AP, then they can be taken as (a – 2d), (a – d), a, (a + d), (a + 2d).