Question

Find the sum of the series 9+3+1+........+1/27​

Answer 1

Answer:

364/27

Step-by-step explanation:

9, 3, 1, ... is a geometric sequence with common ratio r = 1/3.

 

The nth term is a1rn-1.

 

If a1rn-1 = 1/27, then 9(1/3)n-1 = 1/27

 

                                32(3-1)n-1 = 3-3

 

                                3231-n = 3-3

 

                                    33-n = 3-3

 

                                      3-n = -3

 

                                         n = 6

 

The sum, Sn, of the first n terms of a geometric sequence is given by Sn = a1(1 - rn) / (1 - r).

 

So, 9 + 3 + 1 + ... + 1/27 = S6 = 9(1 - (1/3)6) / (1 - 1/3)

 

                                              = 9(1 - 1/729) / (2/3)

 

                                              = (27/2)(728/729) = 364/27