\(D=\frac{2.2012}{1+\frac{2}{2.\left(1+2\right)}+\frac{2}{2\left(1+2+3\right)}+\frac{2}{2\left(1+2+3+4\right)}+...+\frac{2}{2\left(1+2+..+2012\right)}}\)
\(=\frac{2.2012}{1+\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+...+\frac{2}{4050156}}\)
\(=\frac{2.2012}{1+2\left(\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{4050156}\right)}\)
\(=\frac{2.2012}{1+2\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{2012.2013}\right)}\)
\(=\frac{2.2012}{1+2.\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{2012}-\frac{1}{2013}\right)}\)
\(=\frac{2.2012}{1+2.\left(\frac{1}{2}-\frac{1}{2013}\right)}\)
\(=\frac{2.2012}{1+\frac{2.2011}{2.2013}}\)
\(=\frac{2.2012}{1+\frac{2011}{2013}}\)
\(=\frac{4024}{\frac{4024}{2013}}\)
\(=2013\)
Vậy D=2013
\(D=\frac{2\cdot2012}{1+\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+...+\frac{1}{1+2+3+...+2012}}\)
\(D=\frac{2.2012}{1+\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{1}{2025078}}\)
\(D=\frac{4024}{1+2\cdot\left(\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{4050156}\right)}\)
\(D=\frac{4024}{2\cdot\left(\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+...+\frac{1}{2012\cdot2013}\right)}\)
\(D=\frac{4024}{1+2\cdot\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2012}-\frac{1}{2013}\right)}\)
\(D=\frac{4024}{1+2\cdot\left(\frac{1}{2}-\frac{1}{2013}\right)}\)
\(D=\frac{4024}{1+2.\left(\frac{2013-2}{4026}\right)}\)
\(D=\frac{4024}{1+2\cdot\frac{2011}{4026}}\)
\(D=\frac{4024}{1+\frac{2011}{2013}}\)
\(D=\frac{4024}{\frac{4024}{2013}}\)
\(D=\frac{1}{2013}\)