\(S=4^{10}+4^{11}+4^{12}+...+4^{2019}\)
\(4S=4\left(4^{10}+4^{11}+4^{12}+...+4^{2019}\right)\)
\(4S=4^{11}+4^{12}+4^{13}+...+4^{2020}\)
\(4S-S=\left(4^{11}+4^{12}+4^{13}+...+4^{2020}\right)-\left(4^{10}-4^{11}+4^{12}+...+4^{2019}\right)\)
\(3S=4^{2020}-4^{10}\)
\(S=\dfrac{4^{2020}-4^{10}}{3}\)