Geometric Sequences 
                                                                                                      
                        Another simple way of generating a sequence is to start with a number “a” and repeatedly 
                        multiply it by a fixed nonzero constant “r”.  This type of sequence is called a geometric 
                        sequence. 
                         
                                    Definition:  A geometric sequence is a sequence of the form 
                                     
                                                                                                 234
                                                                                    aa,   r, ar, ar, ar,... 
                                                                                                      
                                             The number a is the first term, and r is the common ratio of the sequence.  The 
                                             nth term of a geometric sequence is given by 
                                              
                                                                                                        n−1
                                                                                             aa= r. 
                                                                                               n
                                                                                                      
                        The number r is called the common ratio because any two consecutive terms of the sequence 
                        differ by a multiple of r, and it is found by dividing any term a                                        after the first by the preceding 
                                                                                                                            n+1
                        term a .  That is 
                                   n
                         
                                                                                                    a
                                                                                              r = n+1 . 
                                                                                                      a
                                                                                                       n
                                                                                                      
                        Is the Sequence Geometric? 
                         
                        Example 1:  Determine whether the sequence is geometric.  If it is geometric, find the common 
                                             ratio. 
                         
                                             (a) 2,8,32,128,... 
                                             (b) 1, 2, 3, 5, 8, ... 
                                              
                                    Solution (a):   In order for a sequence to be geometric, the ratio of any term to the  
                                                            one that precedes it should be the same for all terms.  If they are all  
                                                            the same, then r, the common difference, is that value. 
                                     
                                    Step 1:  First, calculate the ratios between each term and the one that precedes it. 
                                     
                                                                                                   8 = 4
                                                                                                   2
                                                                                                 32 = 4 
                                                                                                  8
                                                                                               128 = 4
                                                                                                32
                         
                                                                                                                                                                By: Crystal Hull 
                Example 1 (Continued): 
                 
                        Step 2:   Now, compare the ratios.  Since the ratio between each term and the one  
                           that precedes it is 4 for all the terms, the sequence is geometric, and the  
                           common ratio r=4. 
                         
                        Solution (b): 
                         
                        Step 1:   Calculate the ratios between each term and the one that precedes it. 
                         
                                                                 2 =1
                                                                 1
                                                                 33
                                                                   =
                                                                 22
                                                                        
                                                                 55
                                                                   =
                                                                 33
                                                                 88
                                                                   =
                                                                 55
                                                                    
                        Step 2:   Compare the ratios.  Since they are not all the same, the sequence is not  
                           geometric. 
                               
                Similar to an arithmetic sequence, a geometric sequence is determined completely by the first 
                term a, and the common ratio r.  Thus, if we know the first two terms of a geometric sequence, 
                then we can find the equation for the nth term. 
                 
                Finding the Terms of a Geometric Sequence: 
                                                                             th
                Example 2:  Find the nth term, the fifth term, and the 100  term, of the geometric sequence 
                              determined by              1
                                              ar==6,       . 
                                                         3
                 
                        Solution:  To find a specific term of a geometric sequence, we use the formula  
                                    for finding the nth term. 
                         
                        Step 1:  The nth term of a geometric sequence is given by 
                         
                                                                      n−1
                                                               aa= r 
                                                                n
                 
                           So, to find the nth term, substitute the given values            1 into the formula. 
                                                                                  ar6,
                                                                                    =     = 3
                            
                                                                     1 n−1
                                                             a =6⎛⎞ 
                                                              n    ⎜⎟
                                                                     3
                                                                   ⎝⎠
                                                                                                           By: Crystal Hull 
                   Example 2 (Continued): 
                    
                            Step 2:   Now, to find the fifth term, substitute n = 5 into the equation for the nth  
                               term. 
                             
                                                                                1 51−
                                                                      a =6⎛⎞
                                                                        5     ⎜⎟
                                                                                3
                                                                              ⎝⎠
                                                                                1
                                                                          =6⎛⎞
                                                                              ⎜⎟
                                                                                 4
                                                                                3
                                                                              ⎝⎠
                                                                                       
                                                                          = 6
                                                                            81
                                                                          = 2
                                                                            27
                                                                               
                                                                               
                            Step 3:  Finally, find the 100th term in the same way as the fifth term. 
                             
                                                                               1 100−1
                                                                      a =6⎛⎞
                                                                       5     ⎜⎟
                                                                               3
                                                                             ⎝⎠
                                                                                1
                                                                         =6⎛⎞
                                                                             ⎜⎟
                                                                                99
                                                                               3
                                                                             ⎝⎠
                                                                                        
                                                                         = 23⋅
                                                                             99
                                                                            3
                                                                         = 2
                                                                             98
                                                                            3
                             
                             
                   Example 3:  Find the common ratio, the fifth term and the nth term of the geometric sequence. 
                    
                                   (a) −−1, 9,  81, 729,... 
                                    
                                                23
                                        1 tt t
                                   (b)  2, 6, 18, 54,... 
                                    
                                    
                            Solution (a):   In order to find the nth term, we will first have to determine what a  
                                              and r are.  We will then use the formula for finding the nth term of  
                                              a geometric sequence. 
                             
                    
                    
                                                                                                                            By: Crystal Hull 
                Example 3 (Continued): 
                 
                 
                        Step 1:   First, determine what a and r are.  The number a is always the first term  
                           of the sequence, so 
                         
                                                                  a = −1. 
                               
                                   The ratio between any term and the one that precedes it should be the same 
                                   because the sequence is geometric, so we can choose any pair to find the 
                                                  r.  If we choose the first two terms 
                                   common ratio 
                            
                                                                r = 9
                                                                    −  
                                                                      1
                                                                  =−
                                                                      9.
                               
                        Step 2:   Since we are given the fourth term, we can multiply it by the common  
                           ratio r=−9 to get the fifth term. 
                         
                                                                =    ⋅
                                                             aar
                                                              54
                                                                =729 −9  
                                                                      ()
                                                                =−6561
                         
                        Step 3:   Now, to find the nth term, substitute ar= −=1,     −9  into the formula for  
                           the nth term of a geometric sequence. 
                         
                                                                   n−1
                                                           aa= r
                                                             n
                                                              =− −n−1 
                                                                   19
                                                                ()()
                                                                        −
                                                              =−− n1
                                                                     9
                                                                  ()
                         
                                                                                                            By: Crystal Hull