Question

Cum se calculeaza
1supra1·2+1supra2·3+...+1supra2012·2013

Answer 1

1/1*2+1/2*3+1/ 3*4+...+1/2011*2012+1/2012*2013=

2-1/1*2+2-3/2*3+4-3/3*4+.........+2012-2011/2011*2012+2013-2012/2012*2013=

Le scriu unele sub altele ca sa fie mai clar

2/1*2-1/1*2+
2/2*3-3/2*3+
4/3*4-3/3*4+
.................
2012/2011*2012-2011/2011*2012+
2013/2012*2013-2012/2012*2013=

1/1-1/2+1/2-1/3+1/3+.........+1/2011-1/2012+1/2012-1/2013=

1-1/2013=
2013-1/2013=
2012/2013

Answer 2

$\frac{1}{1*2} + \frac{1}{2*3}+ \frac{1}{3*4}+...+ \frac{1}{2012*2013}=$
$=(\frac{1}{1}- \frac{1}{2}) + (\frac{1}{2}- \frac{1}{3}) + ( \frac{1}{3}- \frac{1}{4}) +...+ (\frac{1}{2012}- \frac{1}{2013}) = \\ =1-\frac{1}{2013}= \frac{2013-1}{2013} = \frac{2012}{2013}$

deoarece $\frac{1}{a}- \frac{1}{a+1} = \frac{a+1-a}{a*(a+1)} = \frac{1}{a*(a+1)}$