SOLUTION: 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

Algebra ->  Sequences-and-series -> SOLUTION: 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      Log On


   

Question 141070: 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.)
Found 2 solutions by vleith, Edwin McCravy:
Answer by vleith(2983) About Me  (Show Source):
You can put this solution on YOUR website!
Look here for some hints http://en.wikipedia.org/wiki/Arithmetic_progression
S%5Bn%5D+=+n%28+2a%5B1%5D+%2B+%28n-1%29d+%29%2F+2+
Let the number of terms be 2k%2B1
0+=+n%28+2a%5B1%5D+%2B+%282k%2B1-1%29d+%29%2F+2+
0+=+n%28+2a%5B1%5D+%2B+%282k%29d+%29%2F+2+
0+=+n%28+a%5B1%5D+%2B+%28k%29d+%29+
-kd = a[1]
Now the function for the n[th] term of series is
a%5Bn%5D+=+a%5B1%5D+%2B+%28n+-+1%29d
a%5Bn%5D+=+-kd+%2B+%28n+-+1%29d
a%5Bn%5D+=+%28n+-+1+-k%29d
a%5Bk%2B1%5D+++=+%28k+%2B1+-+1+-k%29d
a%5Bk%2B1%5D+++=+%280%29d
a%5Bk%2B1%5D+=+0


Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
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.)

          Sn = 0                       (given)

     Sn = n%2F2[2a1 + (n-1)d]           (formula for sum of the first n terms) 

       n%2F2[2a1 + (n-1)d] = 0            (Both sides equal Sn) 

           2a1 + (n-1)d = 0            (Divide both sides by n%2F2

        2a1 + (2k+1-1)d = 0            (Substitute 2k+1 for n) 

              2a1 + 2kd = 0            (Simplification)

                    2a1 = -2kd         (Subtract 2kd from both sides

                     a1 = -kd          (Divide both sides by 2) 

Now we find ak+1

             an = a1 + (n-1)d          (Formula for the nth term) 

           ak+1 = -kd + (k+1-1)d       (Substituting k+1 for n 
                                        and -kd for a1  
            
           ak+1 = -kd + kd             (Simplification)  

           ak+1 = 0                    (Simplification) 

Edwin