Geometric progression

Question

The second term of a convergent geometric series is 5 and the sum to infinity is -125 divided by 6. find the common ratio of the series.

Answer 1

A geometric series can be written as:  a + ar + ar²+ ...

 

The sum of a convergent geometric series is a/(1 - r)

 

So, we have:  a/(1 - r) = -125/6 and ar = 5

 

Since ar = 5, a = 5/r

 

So, (5/r)/(1 - r) = -125/6

 

5/[r(1 - r)] = -125/6

 

Cross multiply to get:  -125r(1 - r) = 30

 

125r² - 125r - 30 = 0

 

Divide both sides by 5:  25r² - 25r - 6 = 0

 

                                         (5r - 6)(5r + 1) = 0

 

                                                             r = 6/5 or -1/5

 

Since we know that the series converges, -1 < r < 1.

 

So,  r = 6/5 is not possible.

 

Thus, r = -1/5