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

1. 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.

2. 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.

3. If a₁, a₂, a₃.....and b₁, b₂, b₃...are two A.P.’s with common differences d  and d′ respectively  then a₁+b₁, a₂+b₂, a₃+b₃,...is also an A.P. with common difference d+d′ 

4. 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

5. 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.

6. If a₁, a₂, a₃, ……. a_n are in A.P.  then a₁ + a_n = a₂ + a_n-1 = a₃ + 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 a₁, a₂, ...an are n numbers then the arithmetic mean (A) of these numbers is

• The n numbers A₁, A₂......An are said to be A.M’s between the numbers a and b  if a, A₁, A₂,........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)