Question

Calculati :
1 supra 1 * 2 + 1 supra 2 * 3 + 1 supra 3 * 4 + ....+ 1 supra 2013 * 2014 =

Answer 1

Formula:
$\frac{1}{n(n+1)}= \frac{1}{n}- \frac{1}{n+1}$

Aplicata pe exercitiu:

$\frac{1}{1\cdot2} + \frac{1}{2\cdot3}+...+ \frac{1}{2013\cdot2014} = \\ 1- \frac{1}{2}+ \frac{1}{2}- \frac{1}{3}+ \frac{1}{3}- \frac{1}{4}+....+ \frac{1}{2013}- \frac{1}{2014}=1- \frac{1}{2014}= \frac{2013}{2014}$

Answer 2

$\displaystyle \frac{1}{1 \cdot 2} + \frac{1}{2\cdot3} + \frac{1}{3 \cdot 4} +...+ \frac{1}{2013 \cdot 2014} = \\ \\ = \frac{1}{1} - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} +...+ \frac{1}{2013} - \frac{1}{2014} = \\ \\ =1- \frac{1}{2014} = \frac{2014-1}{2014} = \frac{2013}{2014}$