For each product, express as a difference of squares:
(100^2-1^2) - (100^2-2^2) + (100^2-3^2) - (100^2-4^2) + ... + (100^2-49^2) - (100^2-50^2)
Distribute the minus signs, and note all the 100^2 cancel:
-1^2 + 2^2 - 3^3 + 4^2 - ... - 49^2 + 50^2
Now group in pairs again:
(-1^2 + 2^2) + (-3^3 + 4^2) + ... + (-49^2 + 50^2)
Reapply difference of squares factorization and simplify a bit:
1*3 + 1*7 + 1*11 + ... + 1*99
Obviously the multiplication by 1 is trivial, leaving behind a simple arithmetic sequence of 25 terms. The sum is easy to evaluate as 25*(3+99)/2 = 1275.
Solution
