\(A=1-\dfrac{3}{4}+\left(\dfrac{3}{4}\right)^2-...+\left(\dfrac{3}{4}\right)^{2020}-\left(\dfrac{3}{4}\right)^{2021}\)
\(\dfrac{3}{4}A=\dfrac{3}{4}-\left(\dfrac{3}{4}\right)^2+\left(\dfrac{3}{4}\right)^3-...+\left(\dfrac{3}{4}\right)^{2021}-\left(\dfrac{3}{4}\right)^{2022}\)
\(\Rightarrow A+\dfrac{3}{4}A=1-\left(\dfrac{3}{4}\right)^{2022}\)
\(\Leftrightarrow\dfrac{7}{4}A=1-\left(\dfrac{3}{4}\right)^{2022}\)
\(\Rightarrow A=\dfrac{4}{7}-\dfrac{4}{7}\left(\dfrac{3}{4}\right)^{2022}\)
Ta có: \(A-\dfrac{4}{7}-\dfrac{4}{7}\left(\dfrac{3}{4}\right)^{2022}< \dfrac{4}{7}< 1\)
\(0< \dfrac{3}{4}< 1\Rightarrow0< \left(\dfrac{3}{4}\right)^{2022}< 1\Rightarrow0< \dfrac{4}{7}\left(\dfrac{3}{4}\right)^{2022}< \dfrac{4}{7}\Rightarrow A>0\)
\(\Rightarrow0< A< 1\Rightarrow\left[A\right]=0\)
