consider the sequence 1 -2 3 -4, 5,-6... what is the average of the first 200 terms of the sequence,

Summing an Arithmetic Series

To sum up the terms of this arithmetic sequence:

                                a + (a+d) + (a+2d) + (a+3d) + ...
use this formula:
                             
Consider the first series in the express:
                            1+3+5+....200 terms

Here,  

n=199, a=1, d=2

  Sum=99.5*((2+198)*2) 

         = 99.5*200*2

         =99.5*402

         =39800

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Another series is:

-2, -4, -6,....200 terms

Here, d=2, n=200, a=-2, n=200

Sum=100*((2+199)*2) 

        = 100*201*2

         =100*402

         =40200

but, this is negative. 

Hence total sum=39800-40200 =-400

Therefore, the average will be -400/400=-1