\(2x:\left(1+\dfrac{1}{1+2}+\dfrac{1}{1+2+3}+...+\dfrac{1}{1+2+3+...x}\right)=2023\left(1\right)\)
Đặt \(A=\left(1+\dfrac{1}{1+2}+\dfrac{1}{1+2+3}+...+\dfrac{1}{1+2+3+...x}\right)\)
\(\Rightarrow A=\left(1+\dfrac{1}{3}+\dfrac{1}{6}+...+\dfrac{1}{\dfrac{x\left(x+1\right)}{2}}\right)\)
\(\Rightarrow\dfrac{1}{2}A=\left(\dfrac{1}{2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{x\left(x+1\right)}\right)\)
\(\Rightarrow\dfrac{1}{2}A=\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{x}-\dfrac{1}{x+1}\right)\)
\(\Rightarrow\dfrac{1}{2}A=1-\dfrac{1}{x+1}\)
\(\Rightarrow A=2\left(1-\dfrac{1}{x+1}\right)\Rightarrow A=\dfrac{2x}{x+1}\)
\(\left(1\right)\Rightarrow2x:\dfrac{2x}{x+1}=2023\)
\(\Rightarrow2x.\dfrac{x+1}{2x}=2023\left(x\ne0\right)\)
\(\Rightarrow x+1=2023\)
\(\Rightarrow x=2022\)
