Question

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Answer 1

Trebuie sa ne folosim de o formula:
$\frac{1}{a(a+k)}= \frac{1}{k} ( \frac{1}{a} - \frac{1}{a+k} )$

[tex]S=(( \frac{1}{2}( \frac{1}{1} - \frac{1}{3} )+ \frac{1}{2}( \frac{1}{3} - \frac{1}{5} ))+...+ \frac{1}{2}( \frac{1}{2009} - \frac{1}{2011} ) )*4022\\\\ S=( \frac{1}{2}(1- \frac{1}{3} + \frac{1}{3} - \frac{1}{5} + \frac{1}{5} -...- \frac{1}{2009} + \frac{1}{2009}- \frac{1}{2011} ) )*4022[/tex]

Dupa cum vezi, se reduc termenii: 1/3 cu -1/3, 1/5 cu -1/5, si tot asa pana la 1/2009 cu -1/2009, si ne ramane 1 - 1/2011:

$S= \frac{1}{2} (1- \frac{1}{2011} )*4022= \frac{2010*2011}{2011}=2010$