<PRE><B>Question 141070</B>
Show that if the sum of an arithmetic series with an odd number of terms is 0, then one of the terms of the series must be 0. (Hint: let the number of terms be 2k + 1. Show that a1 = -kd. Then find the (k + 1)st term.)
&lt;pre&gt;&lt;font size = 4 color = &quot;indigo&quot;&gt;&lt;b&gt;
          S&lt;sub&gt;n&lt;/sub&gt; = 0                       (given)

     S&lt;sub&gt;n&lt;/sub&gt; = {{{n/2}}}[2a&lt;sub&gt;1&lt;/sub&gt; + (n-1)d]           (formula for sum of the first n terms) 

       {{{n/2}}}[2a&lt;sub&gt;1&lt;/sub&gt; + (n-1)d] = 0            (Both sides equal Sn) 

           2a&lt;sub&gt;1&lt;/sub&gt; + (n-1)d = 0            (Divide both sides by {{{n/2}}}

        2a&lt;sub&gt;1&lt;/sub&gt; + (2k+1-1)d = 0            (Substitute 2k+1 for n) 

              2a&lt;sub&gt;1&lt;/sub&gt; + 2kd = 0            (Simplification)

                    2a&lt;sub&gt;1&lt;/sub&gt; = -2kd         (Subtract 2kd from both sides

                     a&lt;sub&gt;1&lt;/sub&gt; = -kd          (Divide both sides by 2) 

Now we find a&lt;sub&gt;k+1&lt;/sub&gt;

             a&lt;sub&gt;n&lt;/sub&gt; = a&lt;sub&gt;1&lt;/sub&gt; + (n-1)d          (Formula for the nth term) 

           a&lt;sub&gt;k+1&lt;/sub&gt; = -kd + (k+1-1)d       (Substituting k+1 for n 
                                        and -kd for a1  
            
           a&lt;sub&gt;k+1&lt;/sub&gt; = -kd + kd             (Simplification)  

           a&lt;sub&gt;k+1&lt;/sub&gt; = 0                    (Simplification) 

Edwin&lt;/pre&gt;</PRE>