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Đặt:\(A=1+\frac{1}{2}.\left(1+2\right)+\frac{1}{3}.\left(1+2+3\right)+......+\frac{1}{20}.\left(1+2+3+....+20\right)\)
\(\Rightarrow A=1+\frac{\left(1+2\right).2}{2.2}+\frac{\left(1+3\right).3}{3.2}+.........+\frac{\left(1+20\right).20}{20.2}\)
\(\Rightarrow A=\frac{2}{2}+\frac{3.2}{2.2}+\frac{4.3}{3.2}+.................+\frac{21.20}{20.2}\)
\(\Rightarrow A=\frac{2}{2}+\frac{3}{2}+\frac{4}{2}+.......+\frac{21}{2}\)
\(\Rightarrow A=\frac{1}{2}.\left(2+3+4+...+21\right)=\frac{1}{2}.\frac{\left(2+21\right).20}{2}=\frac{1}{2}.\frac{23.20}{2}=\frac{23.2.2.5}{2.2}=23.5=115\)
Vậy A=115
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