## Difference of Sequences

For the sequence $1,-2,3,-4,5,-6,7,\cdots$ , what is the difference between the mean of the sequence’s first $400$ terms and the mean of its first $200$ terms?
Source: NCTM Mathematics Teacher February 2008

Solution
If we add the first $400$ terms by pairs $1+(-2),3+(-4),5+(-6),\cdots$ we end up with $200(-1)=-200$. The mean of the first $400$ terms is $-200/400=-1/2$.
Likewise, if we add the first $200$ terms by pairs, we get $100(-1)=-100$. The mean of the first $200$ terms is $-100/200=-1/2$.
The difference between the two means is $-1/2-(-1/2)=0$.

Answer: $0$