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.

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