# Arithmetic Progression (AP)

If

*a*is the first term and*d*is the common difference, then AP can be written as*a*,

*a*+

*d*,

*a*+ 2

*d*, …,

*a*+ (

*n*– 1)

*d*, …

*n***th term:**

*T*

_{n}*= a*+ (

*n*– 1)

*d*=

*l*(last term), where

*d*=

*T*–

_{n}*T*

_{n}_{–1}.

*n***th term from last:**

*T*′ =

_{n}*l*– (

*n*– 1)

*d*.

# Sum of *n* terms of an AP

The sum

*S*of_{n}*n*terms of an AP with first term “*a*” and common difference “*d*” is*S*= [2

_{n}*a*+ (

*n*– 1)

*d*] or

*S*= [

_{n}*a*+

*l*]

where

*l*= last term =*a*+ (*n*– 1)*d.*Thus, in general, sum of

*n*terms of AP is*S*=_{n}*An*^{2}+*Bn*.# Some important facts about AP

- If a fixed number is added or subtracted to each term of a given AP, then the resulting sequence is also an AP, and its common difference remains same.
- If each term of an AP is multiplied by a fixed constant or divided by a fixed non-zero constant, then the resulting sequence is also an AP.
- If
*x*_{1},*x*_{2},*x*_{3}, … and*y*_{1},*y*_{2},*y*_{3}, …, are two AP’s then*x*_{1}±*y*_{1},*x*_{2}±*y*_{2},*x*_{3}±*y*_{3}+ … are also AP’s - Three terms in AP should preferably be taken as
*a*–*d*,*a*,*a*+*d*and four terms as*a*– 3*d*,*a*–*d*,*a*+*d*,*a*+ 3*d*. - In AP,
*a*= (1/2) (_{n}*a*_{n}_{ – k}+*a*_{n}_{ + k}), for*k*≤*n*. - In AP,
*a*_{1}+*a*=_{n}*a*_{2}+*a*_{n}_{–1}=*a*_{3}+*a*_{n}_{–2}= …

# Insertion of Arithmetic Means

If between two given numbers

*a*and*b*we have to insert*n*numbers*A*_{1},*A*_{2}, …,*A*such that_{n}*a*,*A*_{1},*A*_{2}, …*A*,_{n}*b*form an AP, then we say that*A*_{1},*A*_{2}, …,*A*are arithmetic means between_{n}*a*and*b*.

*Notes:**r*th mean is*A*= , where_{r}*r*= 1, 2, 3, …,*n.*- The sum of
*n*arithmetic means between two numbers is*n*times the single AM between them or*A*_{1}+*A*_{2}+ … +*A*= (AM between_{n}*a*and*b*).