Geometric Sequence and Series

 

Motivation:

 

                       A geometric sequence is with the following relationship

                                          A, Ar, Ar², Ar³, Ar⁴, …

 

                       In other words,  each term is given by a multiple r of the previous term. Geometric sequence and series are widely used in financial application. And it also has many good properties that can be used in engineering design.

 

 

 

 

 

 

Definition ( Geometric Sequence )

                          The following sequence is known as geometric sequence

                                           A, Ar, Ar², Ar³, Ar⁴, …

 

                          where A is the first term and r is known as “common ratio”.

 

 

 

 

 

 

Geometric Series

                      Let

                                   S_n = A + Ar + Ar² + Ar³ + … + Ar^n-1  .

 

                      We multiply this equality by r :

 

                                 rS_n   =        Ar+ Ar² + Ar³ + … + Ar^n-1 + Arⁿ   

 

                      Then we subtract the two equation:

 

                                 S_n – rS_n = A - Arⁿ

 

                      So,

                                                            S_n =

 

                      If  we subtract with the other way around, we get

                                               rS_n – S_n  =  Arⁿ – A

                                    S_n =

 

                        The two results   S_n =   and S_n =  are the same. 

 

 

Example:     1+ 2 + 2² + 2³ + … + 2⁹⁹ = ?

                          There are 100 terms of it. Thus,

                                 S =  = 2¹⁰⁰ -1

 

                     It has the property that the next term minus 1 is equal to sum from the first term to the current term.