Introduction to Geometric Progression

GEOMETRIC PROGRESSION(G.P.):

⮚A progression is called a G.P. if the ratio of it's each term to it's previous term is always constant. If       'a'  be the first term and 'r' be the common ratio then a, ar, ar², ... ,ar^n-1 is a sequence of G.P . t_n =               ar^n-1. 

GENERAL TERM OF A G.P.:

•   n^th term of G.P. i.e.  t_n = ar^n-1. , where the common ratio (r) =  t₂ /  t₁ =  t₃ /  t₂ = ...
• If G.P.  consists of ' n ' terms , then  p^th term from the end = (n-p+1)^th term from the beginning = ar^n-p. Also , the   p^th   term from the end of the G.P. with the last term ' l ' and common ratio 'r'  is =  l(1/r)^n-1 . 
• 
  
  

SUM OF FIRST ' n ' TERMS OF A G.P.:

⮚ Sum of first ' n ' terms of a G.P. is given by

             S_n = ∑   t_n 

                  =   t₁ +   t₂  +   t₃ + ... +   t_n    

                  =   a + ar + ar²+ ... +ar^n-1

•  ^S_n =  a(1-rⁿ )/(1-r)  and  S_n = (a-lr)/(1-r) , [when | r | < 1. ] 
•  ^S_n =  a(rⁿ -1)/(r-1)  and  S_n = (lr-a)/(r-1) , [when | r | > 1. ]
•  S_n = na  [ when r=1 ]

SUM OF INFINITE TERMS OF G.P.:

⮚ When n→∞ and | r | < 1 ( -1 < r < 1),      S_∞  = a/(1-r).

⮚ When n→∞ and | r | ≥ 1 ,  S_∞  does not exist.

GEOMETRIC MEAN(G.M.):

⮚ If a , b , c are in G.P.  , then  b/a = c/b  ⇒ b²= ac ⇒ b= √(ac)  is the G.M. of a and c.

      Similarly  G.M. of a , b , c  is   (abc)^1/3 .

      G.M.  of   a₁ , a₂  , a₃ , ... , a_n  is     (a₁. a₂  . a₃  ...  .a_n )^1/n         

  ⮚ Let n G.M.s are inserted between a & b . Let    G₁ , G₂  , G₃ , ... , G_n   are n G.M.s ,

      then a ,  G₁ , G₂ , G₃ , ... , G_n , b are in G.P. .

       Then common ratio ( r ) =  (b/a )^1/n+1          

         ^So,   G₁  =   ar   =   a . (b/a )^1/n+1 

                  G₂  =   ar²  =   a . (b/a )^2/n+1 

                  ^{___________________________}

                 G_n  =   arⁿ =   a . (b/a )^n/n+1 

PROPERTIES OF G.P. :

✪  If all the terms of a G.P. be multiplied or divided by the same non-zero constant , then it                 remains a G.P.  , with the same common ratio ' r '.  

✪  The reciprocal of the terms of a given G.P.  form a G.P. with common ratio as reciprocal of the         common ratio(r) of the original G.P. i.e. ' 1/r '.

✪  If each term of a G.P.  with common ratio ' r ' be raised to the same power k , the resulting              sequence also forms a G.P. with common ratio  r^k  .

✪  In a finite G.P. , the product of terms equidistant from the beginning and the end is always the       same and is equal to the product of the first and last term i.e. if   a₁ , a₂  , a₃ , ... , a_n   be in G.P.        . Then                a₁. a_n  = a₂ . a_n-1  =  a₃ . a_n-2  = ... = a_r . a_n-r+1          

✪  If  a₁ , a₂  , a₃ , ... , a_n , ... is a G.P. of non-zero , non-negative terms , then  loga₁ , loga₂  ,log a₃          , ... ,  loga_n , ...  is in A.P. and vice-versa.

✪  Three non-zero numbers a , b , c are in G.P. iff  b²= ac .

✪  If    a^x1 , a^x2, a^x3, ... , a^xn   are in G.P. , then  x₁ , x₂  , x₃ , ... , x_n are in A.P. .

✪  If the first term of a G.P. of n terms is ' a ' and last term is ' l ' , then the product of all terms         of  the G.P.  is  (al )^n/2 .

  ⮚ So, in the below tables , these are terms taken , which should be used in some kinds of questions in        G.P. as same as in the  A.P. .

                  TABLE-1 : When the product is given.

            ^{No. of terms                        Terms taken} 

3	a/r , a , ar
4	a/ r3 , a/r , a/r , a/ r3
5	a/ r2 , a/r , a , a/r , a/ r2

                   TABLE -2 : When the product is not given. 

       

         ^{No. of terms                        Terms taken} 

3	a , ar , ar2
4	a , ar , ar2 ,  ar3
5	a , ar , ar2, ar3, ar4

       So guys , this is all about the introduction of G.P. .      And on later, we will learn about                         Harmonic  Progression and it's properties . 

       Thanks for reading this post .😇😇😇 

                                     And Stay tuned guys . ✌✌✌

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.😇