What is 100^2-99^2+98^2-97^2+96^2-95^2 +...+ 4^2-3^2+2^2-1^2 ?
(Find an expression where n is the largest number being squared, then use it to calculate the sum when n = 100. Hint: Examine specific cases where n is small. Then generalize).
The sum from 1 to n where n is even (2k) is
(2^2-1^2) + ... + ((2k)^2-(2k-1)^2)
= (2+1)(2-1) + (4+3)(4-3) + ... + (2k+2k-1)(2k-2k+1)
= 3+7+...+4k-1
= k/2 (3+4k-1)
= k(2k+1)