Question

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Aratati ca 1/1·2+1/2·3+...+1/2011·2012<2

Answer 1

$\displaystyle \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} +...+ \frac{1}{2011 \cdot 2012}\ \textless \ 2 \\ \\ \frac{1}{1} - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4}+...+ \frac{1}{2011} - \frac{1}{2012} \ \textless \ 2 \\ \\ 1- \frac{1}{2012} \ \textless \ 2 \\ \\ \frac{2012-1}{2012} \ \textless \ 2 \\ \\ \frac{2011}{2012} \ \textless \ 2$