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Arithmetic Progression (AP)

If a is the first term and d is the common difference, then AP can be written as
a, a + d, a + 2d, …, a + (n – 1) d, …
nth term: Tn = a + (n – 1) d = l (last term), where d = TnTn–1.
nth term from last: Tn′ = l – (n – 1)d.

Sum of n terms of an AP

The sum Sn of n terms of an AP with first term “a” and common difference “d” is
Sn = 63193.png [2a + (n – 1)d] or Sn = 63187.png [a + l]
where l = last term = a + (n – 1)d.
Thus, in general, sum of n terms of AP is Sn = An2 + Bn.

Some important facts about AP

  1. If a fixed number is added or subtracted to each term of a given AP, then the resulting sequence is also an AP, and its common difference remains same.
  2. If each term of an AP is multiplied by a fixed constant or divided by a fixed non-zero constant, then the resulting sequence is also an AP.
  3. If x1, x2, x3, … and y1, y2, y3, …, are two AP’s then x1 ± y1, x2 ± y2, x3 ± y3 + … are also AP’s
  4. Three terms in AP should preferably be taken as ad, a, a + d and four terms as a – 3d, ad, a + d, a + 3d.
  5. In AP, an= (1/2) (ank + an + k), for kn.
  6. In AP, a1 + an = a2 + an–1 = a3 + an–2 =

Insertion of Arithmetic Means

If between two given numbers a and b we have to insert n numbers A1, A2, …, An such that a, A1, A2, … An, b form an AP, then we say that A1, A2, …, An are arithmetic means between a and b.


  • rth mean is Ar = 65997.png, where r = 1, 2, 3, …, n.
  • The sum of n arithmetic means between two numbers is n times the single AM between them or A1 + A2 + … + An = 65991.png (AM between a and b).

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