Dam factor comun pe x si obtinem:
[tex]x( \frac{1}{1*2}+ \frac{1}{2*3}+...+ \frac{1}{2009*2010}+ \frac{1}{2010*2011})=2010 \ \textless \ =\ \textgreater \ \\ \ \textless \ =\ \textgreater \ x( 1-\frac{1}{2}+ \frac{1}{2}- \frac{1}{3}+...+ \frac{1}{2009}- \frac{1}{2010} + \frac{1}{2010}- \frac{1}{2011})=2010\ \textless \ =\ \textgreater \ \\ \ \textless \ =\ \textgreater \ x(1- \frac{1}{2011} )=2010\ \textless \ =\ \textgreater \ \\ \ \textless \ =\ \textgreater \ x* \frac{2010}{2011}=2010=\ \textgreater \ \boxed{x=2011} . \\ \\ Solutie:~x=2011.[/tex]
*Am folosit identitatea [tex] \frac{1}{k(k+1)}= \frac{1}{k}- \frac{1}{k+1} [/tex] care se verifica astfel:
[tex] \frac{1}{k}- \frac{1}{k+1}= \frac{k+1}{k(k+1)}- \frac{k}{k(k+1)}= \frac{1}{k(k+1)} [/tex].
