<PRE><B>Question 740508</B>
&lt;pre&gt;
S&lt;sub&gt;4&lt;/sub&gt;= -3.5, S&lt;sub&gt;5&lt;/sub&gt;= -3.75, S&lt;sub&gt;6&lt;/sub&gt;= -3.875.

The sum of the first 5 terms S&lt;sub&gt;5&lt;/sub&gt; minus the sum
of the first 4 terms S&lt;sub&gt;4&lt;/sub&gt; is the fifth term a&lt;sub&gt;5&lt;/sub&gt;.  
That is:

-3.75 = S&lt;sub&gt;5&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt;+a&lt;sub&gt;2&lt;/sub&gt;+a&lt;sub&gt;3&lt;/sub&gt;+a&lt;sub&gt;4&lt;/sub&gt;+a&lt;sub&gt;5&lt;/sub&gt;
-3.5  = S&lt;sub&gt;4&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt;+a&lt;sub&gt;2&lt;/sub&gt;+a&lt;sub&gt;3&lt;/sub&gt;+a&lt;sub&gt;4&lt;/sub&gt; 
-----------------------------------

Subtract those equations term by term

-3.75 - (-3.5) = a&lt;sub&gt;5&lt;/sub&gt;  
   -3.75 + 3.5 = a&lt;sub&gt;5&lt;/sub&gt;
         -0.25 = a&lt;sub&gt;5&lt;/sub&gt;

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Similarly, the sum of the first 6 terms S&lt;sub&gt;6&lt;/sub&gt; 
minus the sum of the first 5 terms S&lt;sub&gt;5&lt;/sub&gt; is the 
sixth term a&lt;sub&gt;6&lt;/sub&gt;. That is:

-3.875 = S&lt;sub&gt;6&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt;+a&lt;sub&gt;2&lt;/sub&gt;+a&lt;sub&gt;3&lt;/sub&gt;+a&lt;sub&gt;4&lt;/sub&gt;+a&lt;sub&gt;5&lt;/sub&gt;+a&lt;sub&gt;6&lt;/sub&gt;
-3.75  = S&lt;sub&gt;5&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt;+a&lt;sub&gt;2&lt;/sub&gt;+a&lt;sub&gt;3&lt;/sub&gt;+a&lt;sub&gt;4&lt;/sub&gt;+a&lt;sub&gt;5&lt;/sub&gt; 
-----------------------------------

Subtract those equations term by term

-3.875 - (-3.75) = a&lt;sub&gt;6&lt;/sub&gt;  
   -3.875 + 3.75 = a&lt;sub&gt;6&lt;/sub&gt;
          -0.125 = a&lt;sub&gt;6&lt;/sub&gt;

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So we have found two consecutive terms of the geometric series.

a&lt;sub&gt;5&lt;/sub&gt; = -0.25
a&lt;sub&gt;6&lt;/sub&gt; = -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 = {{{(-0.125)/(-0.25)}}} = 0.5 
 
Edwin&lt;/pre&gt;</PRE>