\(E=\frac{1}{2}\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{10.11.12}\right)\)
\(=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-...-\frac{1}{10.11}+\frac{1}{10.11}-\frac{1}{11.12}\right)\)
\(=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{11.12}\right)=\frac{1}{2}.\frac{65}{132}=\frac{65}{264}\)
Ta có :
\(E=\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{10.11.12}\)
\(2E=\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{10.11.12}\)
\(2E=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{10.11}-\frac{1}{11.12}\)
\(2E=\frac{1}{1.2}-\frac{1}{10.11}\)
\(2E=\frac{1}{2}-\frac{1}{110}\)
\(2E=\frac{27}{55}\)
\(E=\frac{27}{55}:2\)
\(E=\frac{27}{55}.\frac{1}{2}\)
\(E=\frac{27}{110}\)
Vậy \(E=\frac{27}{110}\)
Chúc bạn học tốt ~
