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.Com
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)   (Show Source): You can put this solution on YOUR website!
Look here for some hints http://en.wikipedia.org/wiki/Arithmetic_progression

Let the number of terms be



-kd = a[1]
Now the function for the n[th] term of series is
Answer by Edwin McCravy(20060)   (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 = [2a1 + (n-1)d]           (formula for sum of the first n terms) 

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

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

        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
RELATED QUESTIONS

Write an arithmetic series with seven terms such that the sum of the series is... (answered by fcabanski,stanbon)
If the sum of the terms of an arithmetic series is 234, and the middle term is 26, find... (answered by jim_thompson5910)
Find the sum of 15 terms of an arithmetic series if the middle term is 92 (answered by greenestamps)
Find the sum of 15 terms of an arithmetic series if the middle term is... (answered by MathLover1)
If there are (2n+1) terms in an arithmetic series, prove that the ratio of the sum of odd (answered by richard1234)
The middle term of an arithmetic series of 31 terms is 50.determine the sum of the... (answered by ikleyn)
The sum of the first 25 terms of an arithmetic series with a common difference of 5 is... (answered by stanbon)
In an arithmetic series the sum of the first 9 terms is 162 and the sum of the first 12... (answered by Theo)
An arithmetic series of 30 375 terms has a sum that is a perfect fifth power. What is the (answered by Edwin McCravy,ikleyn)