Arithmetic progression formula
Lets understand Arithmetic progression formula
An A.P. is a sequence whose terms increase or decrease by a fixed number, called the common difference of the A.P. Check out Maths Formulas and NCERT Solutions for class 12 Maths prepared by Physics Wallah.
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 an is given by an = a + (n  1)d.
The sum S_{n} of the first n terms of such an A.P. is given by (a + l ) where l is the last term (i.e. the nth term of the A.P.).
Importnat pointer about Arithmetic progression formula

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 nonzero 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_{n1} = a_{3} + a_{n –}^{2} = . . . . . and so on.
Arithmetic Mean(s):

If three terms are in A.P., then the middle term is called the arithmetic mean (A.M.) between the other two i.e. if a,b,c are in A.P. then is the A.M. of a and c.

If a_{1}, a_{2}, ...an are n numbers then the arithmetic mean (A) of these numbers is

The n numbers A_{1}, A_{2}......An are said to be A.M’s between the numbers a and b if a, A_{1}, A_{2},........An,b are in A.P. If d is the common difference of this A.P. then
b = a + (n + 2  1)d ⇒
⇒ .
Example: If the I^{st }and the 2^{nd} terms of an A.P are 1 and –3 respectively, find the n^{th} term and the sum of the I^{st} n terms.
Detail Explanation : I^{st} term = a, 2^{nd} term = a + d where a = 1, a + d = 3,
⇒ d = 4 (Common difference of A.P.)
we have an = a + (n –1)d
= 1 + (n – 1) (4) = 5 – 4n
Sn = {a + an} = {1 + 5 – 4n} = n (3 – 2n)