Question

1) Calculati 1/n - 1/n+1 = ?
2) 1/1*2+1/1*3+1/3*4....+1/2007*208+1/2008*2009+1/2009*2010=?

Answer 1

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

$2) \frac{1}{1*2}= \frac{1}{1} - \frac{1}{2} \\ \\ \frac{1}{2*3}= \frac{1}{2}- \frac{1}{3} \\ \\ \frac{1}{3*4}= \frac{1}{3}- \frac{1}{4} \\ ........................................... \\ \frac{1}{2007*2008}= \frac{1}{2007}-\frac{1}{2008} \\ \\ \frac{1}{2008*2009}= \frac{1}{2008}-\frac{1}{2009} \\ \\ \frac{1}{2009*2010}= \frac{1}{2009}-\frac{1}{2010} \\\\\\ S= \frac{1}{1}^{(2010} - \frac{1}{2010} = \frac{2010-1}{2010}\\ \\ \boxed{S= \frac{2009}{2010} }$