Question

Aratati ca : $\frac{1}{2*3} + \frac{1}{3*4} + \frac{1}{4*5} +...+ \frac{1}{2015*2016} \ \textless \ \frac{1}{2}$

Answer 1

1/k*(k+1) =1/k -1/(k+1) =>
=>
(1/2 -1/3)+(1/3-1/4)+....+(1/2005-1/2006)=
=1/2 -1/2006 =1002/2006 <1003/2006=1/2