Question

Aratati ca $\frac{1}{1x2} + \frac{1}{2x3} +....+ \frac{1}{2011x2012} \ \textless \ 1$

Answer 1

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