Geometric progression

Geometric progression

{ 1,3,9,27,...}

 { 2,4,8,16,...}

 { 3,12,48,192,...}

What is the similarly among the above sequences of numbers?

  It is very clear. All the terms of the sequence (excluding the first term) can be obtained easily just by multiplying a constant number with the previous term.

This special type of sequence of numbers is known as Geometric progression (G.P.).

Definition:

A sequence of numbers {u(1), u(2), u(3),...}

 forms  a geometric progression if the value of the ratio, u(n+1)/u(n) is a constant for every positive integer n.

The ratio u(n+1)/u(n) is called the common ratio of the geometric progression.

General form of a G.P.

The most general form of a geometric progression is { a,ar,ar²,ar³,...}, where "a" and "r" are the first term and common ratio of the geometric progression.

The n-th term of a G.P.

If t(n) be the n-th term of a geometric progression whose first term is "a" and common ratio is "r" then,

  t(n)=arⁿ⁻¹.

example:
   find out the 10-th term of the geometric progression {2, 6, 18,...}.
    Here, the first term is 2 and the common ratio is 3.
  So, the 10-th term is= 2×(3)¹⁰⁻¹ =2×3⁹.
   

Sum of first n-terms of a G.P.

If "a"and "r" are  the first term and common ratio of a geometric progression, then the sum of the geometric progression upto first n-terms is:

   Assuming, "r" not equals to 1,

S(n)= a×{(1-rⁿ)/(1-r)}, |r|<1.

and, S(n)= a×{(rⁿ -1)/(r-1)}, r>1 or, r<-1.

    In particular if r=1, S(n) =a+a+...+(upto n terms)= na.

    example:
        find the sum of n terms of the geometric progression {1,3,9,27,...}.
        Here, the first term is 1 and the common ratio is 3(>1).
        So, the sum of n terms is
   = {1×(3ⁿ -1)}/(3-1)
   = (3ⁿ -1)/2.

Geometric mean:

 If x be the Geometric mean of a and b then,  x/a =b/x

      or, x²=ab

      or, x=√(ab) or, -√(ab).

Thus the Geometric mean of two numbers is the square root of their product.
      example:
        In the Geometric progression {2,4,8}, 4 is the geometric mean between 2 and 8.

     In general in a Geometric progression with finite  number of terms, all the terms between the first and last terms are the geometric means between the first and last term.

   example:
     In the geometric progression {1,3,9,27,81}, the terms 3,9 and 27 are the geometric means between 1 and 81.

Note:

(1) If the product of three terms of a geometric progression  is given then we consider the terms as, a/r, a,ar.

(2)If the product of  4 terms of a geometric progression is given , then we consider the terms as, a/r³, a/r, ar, ar³.

(3) If the first term and the common ratio of a geometric progression  is known, we can find any term of the geometric progression.

(4) If we know two terms (consecutive) of a Geometric progression, we can determine the whole geometric progression.

(5) The reciprocal terms of a  geometric progression also forms a geometric progression.

The infinite Geometric series:

A series of the form:

  a+ar+ar³+...+arⁿ+...infinity, 

is called an infinite geometric series.

   As for example, 1+1/2+1/(2²)+...+infinity.

 Sum of an infinite geometric series:

 If , -1<r<1 , then , sum is =a+ar+ar²+...+infinity= a/(1-r).

 In particular, if a=1, sum is= 1+r+r²+...+infinity=1/(1-r).

If you find out any incorrect information or know anything more about this , please write it in the comment section!