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.

nth 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 an is given by an = a + (n – 1)d.

The sum Sn 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, ……. an are in A.P.  then a1 + an = a2 + an-1 = a3 + an –2 = . . . . . and so on.

Illustration 1:      The interior angles of a polygon are in arithmetic progression. The smallest angle is 120o and the common difference is 5o. Find the number of sides of the polygon.  Click Here to Go Back to CBSE Class 11 Maths Sequences and Series All Topic Notes