a, \(\frac{1}{2009}+\frac{2}{2009}+...+\frac{2008}{2009}\\ \frac{\left(1+2008\right)\cdot2008\div2}{2009}=\frac{2017036}{2009}\)

\(2010^2-2009^2+2008^2-...+2^2-1^2\)

\(=-\left(1^2-2^2+3^2-...+2009^2-2010^2\right)\)

\(=-\left[1^2+2^2+...+2009^2+2010^2-\left(2^2+4^2+...+2010^2\right)\right]\)

\(=-\left[\frac{2010.\left(2010-1\right)\left(2.2010-1\right)}{6}-2^2\left(1^2+2^2+...+1005^2\right)\right]\)

\(=-\left[2704847285-2^2.\frac{1005\left(1005-1\right)\left(2.1005-1\right)}{6}\right]\)

\(=-\left(2704847285-1351414120\right)=1353433165\)

 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

\(C=\frac{\frac{1}{2008}-\frac{1}{2009}-\frac{1}{2010}}{\frac{5}{2008}-\frac{5}{2009}-\frac{5}{2010}}+\frac{\frac{2}{2007}-\frac{2}{2008}-\frac{2}{2009}}{\frac{3}{2007}-\frac{3}{2008}-\frac{3}{2009}}\)

\(=\frac{\frac{1}{2008}-\frac{1}{2009}-\frac{1}{2010}}{5.\left(\frac{1}{2008}-\frac{1}{2009}-\frac{1}{2010}\right)}+\frac{2.\left(\frac{1}{2007}-\frac{1}{2008}-\frac{1}{2009}\right)}{3.\left(\frac{1}{2007}-\frac{1}{2008}-\frac{1}{2009}\right)}\)

\(=\frac{1}{5}+\frac{2}{3}\)

\(=\frac{13}{15}\)

Cho cách giải lun

\(\Leftrightarrow\dfrac{x+1}{2010}+1+\dfrac{x+2}{2009}+1+...+\dfrac{x+2009}{2}+1+\dfrac{x+2010}{1}+1=0\)

=>x+2011=0

hay x=-2011

a,S1=1+(-2)+3+(-4)+..........+2009+(-2010)

S1=-1.(2010:2)

S1=-1005

b,S2=1+(-2)+(-3)+4+5+(-6)+(-7)+............+2008+2009+(-2010)

S2=-1.(2010:2)

S2=-1.1005

S2=-1005