Introduction to Sequences and Series (Chapter-1)

PROGRESSION:

» A progression is a sequence whose terms a certain pattern i.e. the terms are arranged  under a definite     rule. 

ARITHMETIC PROGRESSION:

»  A sequence of numbers < t_{n  >}is said to be in arithmetic progression (A.P) when the difference                t_n -  t_n-1  is a constant ∀ n є N. This constant is called the  common difference and is denoted by 'd '.

»  If 'a' is the first term and 'd' is the common difference then the A.P can be represented as 

                                         a ,a+d ,a+2d ,a+3d ,...

GENERAL TERM OF AN A.P:

» nth term of an A.P i.e.  t_n =a+(n-1)d.

SUM OF 'n' TERMS OF AN A.P:

» The sum of 'n' terms of the series is given by 

           S_n   =∑  t_n   =  t₁ + t₂    + t₃  +... +t_n 

                             = n/2 [ 2a+(n-1)d]    

       or,

           S_n   =n/2 [a+l].    [where a=first term  and l= last term]

  » NOTE:        t_n   =   S_n -  S_n-1

  ARITHMETIC MEAN:

 » If three terms are in A.P then the middle term is called A.M between the other two.

    Ex.- a ,a+d ,a+2d are in A.P , then a+d is the A.M of a and (a+2d)

            i.e. a+d = [a+(a+2d)]/2

      🟇 A.M of a,b is = (a+b)/2.

     🟇 A.M of a,b,c is = (a+b+c)/3.

 » If  'n' terms are inserted between a & b i.e.  A₁ ,A₂ ,..., A_n  are A.Ms that means  

     a , A₁ ,A₂ ,..., A_n  , b are in A.P. Then the common difference is(d)=. (b-a)/(n+1). 

         Then  A₁ = a + d = a+(b-a)/(n+1).

                   A₂ = a + 2d = a+ 2[(b-a)/(n+1)].

                   --------------------------------------

                    A_n = a + nd = a+ n[ (b-a)/(n+1)].   

PROPERTIES of A.P:

✮ If a fixed number is added or subtracted i.e.a₁士k , a₂士k , a₃ 士k ,... of given A.P , then the resulting       sequence is also an A.P with the common difference as that of given A.P .

✮ If each term of an A.P is multiplied by a fixed number or divided by non-zero fixed number i.e

      ka₁ , ka₂ , ka₃ ,...  ; a₁/k , a₂/k , a₃ /k ,...[k≠0] , then the resulting sequence is also an A.P . Common          difference is multiplied by the fixed number.

✮ The sum of terms of an A.P equidistant from the beginning and the end is equal to sum of first and         last term .        i.e.   a₁  + a_n    = a₂ + a_n-1 =a₃ + a_n-2   and               so on.

✮ If a₁ , a₂ , a₃ ,... and b₁ , b₂ , b₃ ,... are two A.Ps with common differences d and d' , then 

      a₁+b₁ , a₂+b₂ , a₃+b₃  ,... is also an A.P with common difference d+d'.

✮ The nth term of any sequence is linear expression in n , then the sequence is an A.P with the common difference is the coefficient of "n".

      

         Ex.- Let                t_n= 2n+1.

                      put n=1,   t₁= 2(1)+1=3

                      put n=2,   t₂= 2(2)+1=5               { d =2}

                      put n=3,   t₃= 2(3)+1=7

       Here, the common difference (d)= coefficient of n=2.

✮ The sum of nth terms of any sequence is quadratic in n , then the sequence is an A.P with common difference twice the coefficient of n^2.

        Ex. - Let           S_n= 3n^2 + 2n+1.       

        As we know, t_n =   S_n -  S_n-1

                   _{= (3n^2 + 2n+1) - [3(n-1)^2 + 2(n-1)+1]}

               _{= 3n^2 +2n +1 -(3n^2+3-6n+2n-2+1)}

               =  3n^2 +2n +1 -3n^2 -3 +6n -2n +2-1

               = 6n -1.                                           

         here d = 6 = twice of the coefficient of n^2.

     

  ➤ So, in the below tables , these are the terms taken , which should be used in some kind of questions in A.P . 

 TABLE -1: When the sum is given.

                  

               No. of terms                   Term taken

3	a-d , a , a+d
4	a-3d , a-d , a+d , a+3d
5	a-2d , a-d , a , a+d , a+2d

           TABLE -2: When the sum is not given.

                  

     No. of terms                   Term taken

3	a , a+d , a+2d
4	a , a+d , a+2d , a+3d
5	a , a+d , a+2d , a+3d , a+4d

 

                   Here is the sample question :-                                                                                                 

               So guys, this is all about the introduction of sequence and series and about the A.P.

               On later, we will learn about G.P and it's properties.

                 ∴ Stay tuned, for next session.😇