\(E=1-\dfrac{1}{7}+\dfrac{1}{7^2}-...+\dfrac{1}{7^{2020}}-\dfrac{1}{7^{2021}}\\ \Rightarrow7E=7-1+\dfrac{1}{7}-...+\dfrac{1}{7^{2019}}-\dfrac{1}{7^{2020}}\\ \Rightarrow7E+E=6+\dfrac{1}{7}-...+\dfrac{1}{7^{2019}}-\dfrac{1}{7^{2020}}+1-\dfrac{1}{7}+\dfrac{1}{7^2}-...+\dfrac{1}{7^{2020}}-\dfrac{1}{7^{2021}}\\ \Rightarrow8E=7-\dfrac{1}{7^{2021}}=\dfrac{7^{2022}-1}{7^{2021}}\\ \Rightarrow E=\dfrac{7^{2022}-1}{8\cdot7^{2021}}\)
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