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 . 
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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 . ✌✌✌