Arithmetic Progression (A.P.)

Arithmetic Progression: An arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant. The constant difference is called the Common Difference.

General form of  AP:      a, a+d, a+2d, a+3d……

Example:  

3,5,7,9,11……… is an arithmetic progression with common difference 2

7,5,3,1,-1,-3….. is an arithmetic progression with common difference -2.

Arithmetic series means the sum of the elements of an arithmetic progression.

   n-th term of Arithmetic Progression is given by:

          an = a + (n - 1)d

Here, a_n  is n-th term_,a is 1^st term ,  n (e.g. n=1,2,3…) is total number of term and d is common difference.

      The sum (S_n) of the first n terms of an arithmetic progression is given by :

S_n = n/2 {2a + (n-1)d}

Solved Examples:

Q1. Is the row 1,11,21,31... an arithmetic progression?

Ans: Yes, it is an arithmetic progression. Its first term is 1 and the common difference is 10.

Q2. Find the 12^th term of the AP: 3,7,11,15……

Ans:    an = a + (n - 1)d  =3+(12-1)4 =   47

Q3. Find.Find the sum of the first 10 numbers of this arithmetic series: 1, 11, 21, 31...

Ans: S_n =  n/2 {2a + (n-1)d}   = 10/2 { 2*1 +(10-1)*10} = 5*92 = 460.

Q4. The first five terms of an arithmetic sequence are given: 4,3,2,1,0,… What is the next term in the sequence?

Ans: In any arithmetic sequence, each term is equal to the previous term plus the common difference, the common difference is −1.

Next term = 0+(-1) = -1.