Question

Calculați​ suma:

S= 1/1*2+1/2*3+1/3*4+.....+1/999*1000

Answer 1

  
[tex]\displaystyle\\ S=\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{3\times4}+\cdots+\frac{1}{998\times999}+\frac{1}{999\times1000}=\\\\ = \frac{1}{1} -\underbrace{\frac{1}{2}+\frac{1}{2}} -\underbrace{\frac{1}{3}+\frac{1}{3}} -\underbrace{\frac{1}{4} +\cdot\cdot}\cdot\underbrace{\cdot\cdot+\frac{1}{998}} -\underbrace{\frac{1}{999}+\frac{1}{999}} -\frac{1}{1000}=\\\\\\ =\frac{1}{1} -\frac{1}{1000}= \frac{1000}{1000} -\frac{1}{1000}=\frac{1000-1}{1000} =\boxed{\bf \frac{999}{1000}} [/tex]