SOLUTION: Determine the common ratio of a geometric series that has these partial sums: S4= -3.5, S5= -3.75, S6= -3.875. I am having a lot of problems trying to find the common ratio. Th

S4= -3.5, S5= -3.75, S6= -3.875.

The sum of the first 5 terms S5 minus the sum
of the first 4 terms S4 is the fifth term a5.  
That is:

-3.75 = S5 = a1+a2+a3+a4+a5
-3.5  = S4 = a1+a2+a3+a4 
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Subtract those equations term by term

-3.75 - (-3.5) = a5  
   -3.75 + 3.5 = a5
         -0.25 = a5

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Similarly, the sum of the first 6 terms S6 
minus the sum of the first 5 terms S5 is the 
sixth term a6. That is:

-3.875 = S6 = a1+a2+a3+a4+a5+a6
-3.75  = S5 = a1+a2+a3+a4+a5 
-----------------------------------

Subtract those equations term by term

-3.875 - (-3.75) = a6  
   -3.875 + 3.75 = a6
          -0.125 = a6

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So we have found two consecutive terms of the geometric series.

a5 = -0.25
a6 = -0.125

We can find the common ratio by dividing ANY term by the preceding 
term, so we can find the common ratio r by dividing the 6th term
by the 5th term:

r =  = 0.5 
 
Edwin