Question

1^2-2^2+3^2-4^2+....+99^2-100^2

Answer 1

$\sum\limits_{k=1}^{50} \Big[(2k-1)^2-(2k)^2\Big] = \\ \\ =\sum\limits_{k=1}^{50} (2k-1-2k)(2k-1+2k) = \\ \\ = \sum\limits_{k=1}^{50} (-1)\cdot (4k-1) = -\sum\limits_{k=1}^{50}(4k-1) = \\ \\ = -4\cdot \dfrac{50\cdot 51}{2} +50 = -4\cdot 25\cdot 51 + 50 = -5050$