lets take this:

1-4+9-16+25-...-10000+10201

and seperate it into:

(1+9+25+....+10201) - (4+16+......+10000)

which is just:

(sum of squares of odd numbers) - (sum of squares of even numbers)

which leads to the second solution...& we should sum it up to 101 & 100 respectively as the last terms of the first & second parts are (101)^2 & (100)^2 sp (2n-1)=101 & (2n)=100

(and thats exactly how you get the sum as you get n=51 for the first & n=50 for the second)