Geometric Progressions

 

A geometric progression has positive terms. The sum of the first six terms is nine times the sum of the first three terms. The seventh term is 320. Find

        (i)   the common ratio

        (ii)  the first term

        (iii) the sum of first five terms

 

Solution:

 

           Sum of first 6 terms = 9 ( Sum of first 3 terms)

                                    

                               1 - r⁶ = 9 (1 - r³)

                       r⁶ - 9r³ + 8 = 0

                   (r³ - 8)(r³ - 1) = 0

                      r = 2    or  r = 1 (rejected as G. P. is increasing)

 

          Seventh term,

                              a.2^7-1 = 320

                                 a.2⁶ = 320

                                     a = 5

 

          Sum to first five terms

                                          

                                           = 155

 

A geometric progression has a positive common ratio, and the sixth term is . The sum of the first two terms and the sum of the third and fourth terms are in the ratio 4 : 9. Find the sum of the first six terms.

 

Solution:

 

            (First term + Second term) / (Third term + Fourth term)

                                     

                                                       

                                                  r² = 9/4        since a is not 0, r is not 1

                                                   r = 3/2        since r >0

 

            Sixth term =

                                      a = 1

 

            Sum of first six terms