# Arithmetic Progression (A.P.)

The sequence <

*x*_{1},*x*_{2},*x*_{3}, ....,*x*, .... > is called an Arithmetic progression (A.P.), if_{n}*x*_{2}â€“*x*_{1 }=*x*_{3}â€“*x*_{2 }=*x*_{4}â€“*x*_{3 }= ...... =*x*â€“_{n}*x*_{n}_{â€“1 }= ....... In general (*x*_{n}_{ }â€“*x*_{n}_{â€“1})*= constant,*_{ }*n*âˆˆ N. This constant difference is called Common Difference . If '*a*' is the first term and '*d*' is the common difference, then the A.P. can be written as*a*+ (*a*+*d*) + (*a*+ 2*d*) + (*a*+ 3*d*) + ...... + {*a*+ (*n*â€“ 1)*d*} + .....Note that

*a*,*b*,*c*are in AP â‡”2*b*=*a*+*c***G**

**eneral Term of an AP:**

General term (i.e.

*n*^{th}term) of an AP is given byT

*=*_{n}*a*+ (*n*â€“ 1)*d*

**NOTE :**

- If a sequence has
*n*terms, then its*n*^{th}term is also denoted by which indicate last term - Common difference can be zero, + ve or â€“ ve.
- If there are
*n*terms in an A.P, then*m*^{th}term from end = (*n*â€“*m*+ 1)^{th}term from beginning.

**S**

**um of first**

*n*terms of an AP:The sum of first n terms of an A.P. is given by

S

*= [2*_{n}*a*+ (*n*â€“ 1)*d*] or S*= [*_{n}*a*+ T*]*_{n}or

**S**

**ome standard results:**

(i) Sum of first

*n*natural numbers,=

(ii) Sum of first

*n*odd natural numbers, =

*n*^{2}(iii) Sum of first

*n*even natural numbers, =

*n*(*n*+ 1)(iv) Sum of squares of first

*n*natural numbers,=

(v) Sum of cubes of first

*n*natural numbers, =

# Arithmetic Mean (A.M)

The A.M. between the two given quantities

*a*and*b*is A if*a*, A,*b*are in A.P.*i.e.*A â€“

*a*=

*b*â€“ A â‡’ A =

**Note:**

A.M. of any

*n*positive numbers*a*_{1},*a*_{2}, ..........,*a*is_{n}A =