Question

Consider the sequence
1,-2, 3,-4, 5, -6,.........., n(-1)^(n+1)

What is the average of first 300 terms of the sequence?

Answer 1

rearrange the terms to see that the first 2n terms are
1+3+5+... - 2(1+2+3+...)
= n^2 - 2(n(n+1)/2)
= n^2 - n^2-n
= -n
so, the average is -n/2n = -1/2

or, rearrange to see that the first 2n terms are
(1-2)+(3-4)+...+(n-(n+1))
= -1 + -1 + ...
= -n
average is -n/2n = -1/2