CBSE Class 11 Maths Notes: Sequences & Series - Arithmetic Progression

ARITHMETIC PROGRESSION (A. P.)

An A.P. is a sequence whose terms increase or decrease by a fixed number, called the common difference of the A.P.

n^th Term and Sum of n Terms:

If a is the first term and d the common difference, the A.P. can be written as a, a + d, a + 2d… The nth term a_n is given by a_n = a + (n – 1)d.

The sum S_n of the first n terms of such an A.P. is given by

where l is the last term (i.e. the nth term of the A.P.).

Notes

If a fixed number is added (subtracted) to each term of a given A.P. then the resulting sequence is also an A.P. with the same common difference as that of the given A.P.
If each term of an A.P. is multiplied by a fixed number(say k) (or divided by a non-zero fixed number), the resulting sequence is also an A.P. with the common difference multiplied by k.
If a₁, a₂, a₃…..and b₁, b₂, b₃…are two A.P.’s with common differences d and d’ respectively then a₁+b₁, a₂+b₂, a₃+b₃,…is also an A.P. with common difference d+d’
If we have to take three terms in an A.P., it is convenient to take them as a – d, a,
a + d. In general, we take a – rd, a – (r – 1)d,……a – d, a, a + d,…….a + rd in case we have to take (2r + 1) terms in an A.P
If we have to take four terms, we take a – 3d, a – d, a + d, a + 3d. In general, we take
a – (2r – 1)d, a – (2r – 3)d,….a – d, a + d,…..a + (2r – 1)d, in case we have to take 2r terms in an A.P.
If a₁, a₂, a₃, ……. a_n are in A.P. then a₁ + a_n = a₂ + a_n-1 = a₃ + a_{n –2} = . . . . . and so on.

Illustration 1: The interior angles of a polygon are in arithmetic progression. The smallest angle is 120^o and the common difference is 5^o. Find the number of sides of the polygon.