Question

(1/2×3+1/3×5+...+1/2009×2011)×4022

Answer 1

   
[tex]\displaystyle\\ \text{Folosim formula:}\\\\ \frac{2}{n(n+2)}= \frac{1}{n} - \frac{1}{n+2} \\\\ \left( \frac{1}{1\times3}+\frac{1}{3\times5}+\frac{1}{5\times7}+\cdots+\frac{1}{2009\times2011} \right)\times4022=\\\\ =\left(\frac{1}{1\times3}+\frac{1}{3\times5}+\frac{1}{5\times7}+\cdots+\frac{1}{2009\times2011}\right)\times2\times2011=\\\\ =\left(\frac{2}{1\times3}+\frac{2}{3\times5}+\frac{2}{5\times7}+\cdots+\frac{2}{2009\times2011} \right)\times2011= [/tex]


[tex]\displaystyle\\ =\left(\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\cdots+\frac{1}{2009}-\frac{1}{2011} \right)\times2011=\\\\ \text{Observam ca fractiile se reduc 2 cate 2 si raman prima si ultima.}\\\\ =\left(\frac{1}{1}-\frac{1}{2011} \right)\times2011=\\\\ =\left(\frac{2011}{2011}-\frac{1}{2011} \right)\times2011=\frac{2010}{2011}\times 2011 = \boxed{\bf 2010}[/tex]