Question

[Erza Scarlet]

1/3 + 1/6 + 1/10 + ...+ 1/x * ( x + 1) = 2011/2013

Answer 1

[Die Devil]

\(=\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+........+\frac{2}{x\left(x+1\right)}=\frac{2011}{2013}\)

\(\Rightarrow\frac{2}{2.3}+\frac{2}{3.4}+\frac{2}{4.5}+........+\frac{2}{x.\left(x+1\right)}=\frac{2011}{2013}\)

\(=2.\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+........+\frac{1}{x.\left(x+1\right)}\right)=\frac{2011}{2013}\)

\(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+.........+\frac{1}{x}-\frac{1}{x+1}=\frac{2011}{2013}:2\)

\(=\frac{1}{2}-\frac{1}{x+1}=\frac{2011}{4026}\Rightarrow\frac{1}{x+1}=\frac{1}{2}-\frac{2011}{4026}=\frac{1}{2013}\)

\(\Rightarrow x+1=2013\Rightarrow x=2012\)