\(C=1+3+3^2+3^3+...+3^{11}\)
\(\Rightarrow C=3^0+3^1+3^2+3^3+...+3^{11}\)
\(\Rightarrow3C=3^1+3^2+3^3+3^4+...+3^{12}\)
\(\Rightarrow3C-C=\left(3^1+3^2+3^3+3^4+...+3^{12}\right)-\left(3^0+3^1+3^2+3^3+...+3^{11}\right)\)
\(\Rightarrow2C=3^{12}-3^0\)
\(\Rightarrow C=\frac{3^{12}-1}{2}\)