\(2010^2-2009^2+2008^2-...+2^2-1^2\)
\(=-\left(1^2-2^2+3^2-...+2009^2-2010^2\right)\)
\(=-\left[1^2+2^2+...+2009^2+2010^2-\left(2^2+4^2+...+2010^2\right)\right]\)
\(=-\left[\frac{2010.\left(2010-1\right)\left(2.2010-1\right)}{6}-2^2\left(1^2+2^2+...+1005^2\right)\right]\)
\(=-\left[2704847285-2^2.\frac{1005\left(1005-1\right)\left(2.1005-1\right)}{6}\right]\)
\(=-\left(2704847285-1351414120\right)=1353433165\)
2010×2010 - 2009×2009 +2008×2008-...+2×2-1×1
=2 x 2010 - 2 x 2009 + .......+ 2 x 2 - 2 x 1
=2x(2010-2009+2008-.......+2-1)
=2x[(2010-2019)+......+(2-1)]
=2x ( 1+ 1+....+1)
=2x1005
=2010
