Question

Aflați n € N* 1/1+2 + 1/1+2+3 + 1/1+2+...n=2013/2015

Answer 1

[tex]\frac{1}{1+2}+\frac{1}{1+2+3}+_{\dots}+\frac{1}{1+2+3+_{\dots}+n}=\frac{2013}{2015}\\ \frac{1}{\frac{2\cdot 3}{2}}+\frac{1}{\frac{3\cdot 4}{2}}+_{\dots}+\frac{1}{\frac{n\cdot (n+1)}{2}}=\frac{2013}{2015}\\ \frac{2}{2\cdot 3}+\frac{2}{3\cdot 4}+_{\dots}+\frac{2}{n(n+1)}=\frac{2013}{2015}\\ 2\left(\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+_{\dots}+\frac{1}{n(n+1)}\right)=\frac{2013}{2015}\\ 2\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+_{\dots}+\frac{1}{n}-\frac{1}{n+1}\right)=\frac{2013}{2015}\\ [/tex]
[tex]2\left(\frac{1}{2}-\frac{1}{n+1}\right)=\frac{2013}{2015}\\ 2\cdot \frac{n+1-2}{2(n+1)}=\frac{2013}{2015}\\ \frac{n-1}{n+1}=\frac{2013}{2015}\\ 2015(n-1)=2013(n+1)\\ 2015n-2015=2013n+2013\\ 2n=4028\Rightarrow \boxed{n=2014}[/tex]