Đặt \(A=\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+..............+\frac{1}{1+2+3+...+20}\)
\(\Rightarrow A=\frac{1}{\frac{\left(1+2\right).2}{2}}+\frac{1}{\frac{\left(1+3\right).3}{2}}+\frac{1}{\frac{\left(1+4\right).4}{2}}+.............+\frac{1}{\frac{\left(1+20\right).20}{2}}\)
\(\Rightarrow A=\frac{2}{2.3}+\frac{2}{3.4}+\frac{2}{4.5}+...........+\frac{2}{20.21}\)
\(\Rightarrow A=2.\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+..............+\frac{1}{20.21}\right)\)
\(\Rightarrow A=2.\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+......+\frac{1}{20}-\frac{1}{21}\right)\)
\(\Rightarrow A=2.\left(\frac{1}{2}-\frac{1}{21}\right)=2.\left(\frac{21}{42}-\frac{2}{42}\right)=2.\frac{19}{42}=\frac{19}{21}\)
Vậy \(A=\frac{19}{21}\)
Chúc bn học tốt
