\(\frac{1}{a+b}=\frac{1}{b+c}=\frac{1}{c+c}\Rightarrow\frac{1}{a+b}=\frac{1}{b+c}\Rightarrow a+b=b+c\)
\(\Rightarrow a=c\left(1\right)\)
\(\frac{1}{b+c}=\frac{1}{c+c}\Rightarrow b+c=c+c\Rightarrow c=b\left(2\right)\)
Từ \(\left(1\right)\)và \(\left(2\right)\)\(\Rightarrow a=b=c\)
\(Q=\frac{a+2021b+c}{a+2022b+c}=\frac{a+2021a+a}{a+2022a+a}\)
\(Q=\frac{a.\left(1+2021+1\right)}{a.\left(1+2022+1\right)}=\frac{2023}{2024}\)
Vậy, \(Q=\frac{2023}{2024}\)