Question

Calculeaza 1/2 + 1/6 + 1/12 + 1/20 +....+1/10100

Answer 1

   
[tex]\displaystyle\\ \text{Folosim formula:}~~\boxed{\frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1}}\\\\\\ \frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\cdots+\frac{1}{10100}=\\\\ =\frac{1}{1\times2}+\frac{1}{2\times 3}+\frac{1}{3\times4}+\frac{1}{4\times 5}+\cdots+\frac{1}{100\times 101}=\\\\ =\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\cdots+\frac{1}{100}-\frac{1}{101}=\\\\ =\frac{1}{1}-\frac{1}{101}= \frac{101-1}{101}=\boxed{\bf \frac{100}{101}} [/tex]