Question

Aratati ca 3/2*5+3/5*8+3/8*11+...3/2012*2015=2013/4030 Va rog repede

Answer 1

$\displaystyle\\ \text{Folosim urmatoarea formula:}\\\\ \frac{3}{n(n+3)}=\frac{1}{n}-\frac{1}{n+3}\\\\ \text{Rezolvare:}\\\\ \frac{3}{2\cdot5}+\frac{3}{5\cdot8}+\frac{3}{8\cdot11}+...+\frac{3}{2012\cdot2015}=\\\\ =\frac{1}{2}-\frac{1}{5}+\frac{1}{5}-\frac{1}{8}+\frac{1}{8}-\frac{1}{11}+...+ \frac{1}{2012}-\frac{1}{2015}=\\\\ \text{Se reduc termenii asemenea.}\\\\ =\frac{1}{2}-\frac{1}{2015}=\frac{2015-2}{2\cdot 2015}=\boxed{\bf \frac{2013}{4030}}\\\\ \text{\bf cctd}$