\(B=\left(-7\right)^0+\left(-7\right)^1+...+\left(-7\right)^{2019}\)
\(\Rightarrow-7B=\left(-7\right)^1+\left(-7\right)^2+...+\left(-7\right)^{2020}\)
\(\Rightarrow B-\left(-7B\right)=\left(-7\right)^0-\left(-7\right)^{2020}\)
\(\Rightarrow8B=1-\left(-7\right)^{2020}=1-7^{2020}\)
\(\Rightarrow B=\dfrac{1-7^{2020}}{8}\)
\(B=\left(-7\right)^0+\left(-7\right)^1+\left(-7\right)^2+...+\left(-7\right)^{2018}+\left(-7\right)^{2019}\)
\(-7B=\left(-7\right)^1+\left(-7\right)^2+\left(-7\right)^3+...+\left(-7\right)^{2019}+\left(-7\right)^{2020}\)
\(-7B-B=\)\(\left[\left(-7\right)^1+\left(-7\right)^2+\left(-7\right)^3+...+\left(-7\right)^{2019}+\left(-7\right)^{2020}\right]-\left[\left(-7\right)^0+\left(-7\right)^1+\left(-7\right)^2+...+\left(-7\right)^{2018}+\left(-7\right)^{2019}\right]\)
\(-8B=\left(-7\right)^{2020}-1\)
\(B=\dfrac{7^{2020}-1}{-8}\)