Question

S=2005+2004-2003-2002+2001+2000-1999-....+5+4-3-2+1​

Answer 1

$\displaystyle S = 1-2-3+4+5-6-7+...-2002-2003+2004+2005\\ \\ S = 1+\sum\limits_{k=1}^{501}\Big[-(4k-2)-(4k-1)+4k+(4k+1)\Big]\\ \\ S=1+\sum\limits_{k=1}^{501}\Big(-4k+2-4k+1+4k+4k+1\Big)\\ \\ S =1+\sum\limits_{k=1}^{501}\big(2+1+1\big)\\ \\ S = 1+\sum\limits_{k=1}^{501}4 \\ \\ S = 1+4\cdot 501 \\ \\ S = 2005$