Geometric Progressions

 

 

A geometric progression is a sequence in which each term (except the first term) is derived from the preceding term by the multiplication of a non-zero constant, which is the common ratio. A geometric series is formed by the addition of the terms in a geometric progression.

 

Examples:

1) 3, 6, 9, 12, ...                     first term 3, common ratio 3

2) 4, -8, 16, -32, ...                 first term 4, common ratio -2

 

Let the first term be a and common ratio be r.

 

General form of a G. P.:

              a, ar, ar², ar³, ...

 

nth term of a G. P.=

               ar^n-1

 

Sum to first n terms of a G. P.:

             

             

 

Geometric mean. When x, y and z are consecutive numbers in a G. P.,

                                       

                                y² = xz

                                        

   y is the geometric mean

 

Properties of a G. P. (summary of the abovementioned points)

 

1. nth term is in the form ar^n-1, where a and r are constant

2. is constant for all n (common ratio).

3.

 

 

The next page deals with examples on the application of the above points in solving problems involving geometric progressions.