Question

calculati (-1)la puterea 1+(-1)la puterea 1+2 +(-1)la puterea 1+2+3+........+(-1)la puterea 1+2+3.....+20​

Answer 1

$\displaystyle S = (-1)^1+(-1)^{1+2}+(-1)^{1+2+3}+...+(-1)^{1+2+3+...+20}\\ \\= \sum\limits_{k=1}^{20}(-1)^{\dfrac{k(k+1)}{2}} \\ \\ =\sum\limits_{k=1}^{10}(-1)^{\dfrac{(2k-1)(2k-1+1)}{2}}+\sum\limits_{k=1}^{10}(-1)^{\dfrac{2k(2k+1)}{2}}\\ \\ = \sum\limits_{k=1}^{10}(-1)^{(2k-1)k}}+\sum\limits_{k=1}^{10}(-1)^{k(2k+1)}\\ \\ =[(-1+1)+(-1+1)+\underset{\text{de 5 ori}}{\underbrace{...}}+(-1+1)]+\\ +[(-1+1)+(-1+1)+\underset{\text{de 5 ori}}{\underbrace{...}}+(-1+1)]\\ \\ = 0+0+0+0+...+0\\ \\ =\boxed{0}$