**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 a
_{1}, a_{2}, a_{3}…..and b_{1}, b_{2}, b_{3}…are two A.P.’s with common differences d and d’ respectively then a_{1}+b_{1}, a_{2}+b_{2}, a_{3}+b_{3},…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 a
_{1}, a_{2}, a_{3}, ……. a_{n}are in A.P. then a_{1}+ a_{n}= a_{2}+ a_{n-1}= a_{3}+ 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.*

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