Question

Aratati ca:
$\frac{1}{1 \times 2} + \frac{1}{2 \times 3} + ...... + \frac{1}{2011 \times 2012} < 1$
va rog este urgent!​

Answer 1

$\displaystyle\\\frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1},~pentru~orice~numar~n.\\--------------------------\\\frac{1}{1\cdot2}=\frac{1}{1}-\frac{1}{2}.\\\frac{1}{2\cdot3}=\frac{1}{2}-\frac{1}{3}.\\\frac{1}{3\cdot4}=\frac{1}{3}-\frac{1}{4}.\\.\\.\\.\\\frac{1}{2010\cdot2011}=\frac{1}{2010}-\frac{1}{2011}.\\\frac{1}{2011\cdot2012}=\frac{1}{2011}-\frac{1}{2012}.\\----------(+)--\\1-\frac{1}{2012}~care~este~evident~mai~mic~decat~1,~scazand~ceva~din~1.$