Squares are Cool...
Number Theory
Level
3
Compute 1^{2} - 2^{2} + 3^{2} - 4^{2} + 5^{2} - 6^{2} + 7^{2} - 8^{2} +… + 457^{2} - 458^{2}.
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The answer is -105111.
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1 solution
Break it down into chunks…
(1^2 - 2^2) + (3^2 - 4^2) + (5^2 - 6^2) + … + (457^2 - 458^2)
a^2 - b^2 = (a+b)(a-b).
In ALL cases, a-b = -1. So you are really adding -(a+b) (different a and b in every chunk of expression). -1 + -2 + -3 + -4 + … + -458 = -(1 + 2 + 3 + 4 + … + 458) = -(458 * 459) ÷ 2 = -105111