\(A=1+3+3^2+...+3^{2000}\)
\(3A=3\left(1+3+3^2+...+3^{2000}\right)\)
\(3A=3+3^2+...+3^{2001}\)
\(3A-A=\left(3+3^2+...+3^{2001}\right)-\left(1+3+...+3^{2000}\right)\)
\(2A=3^{2001}-1\). Suy ra
\(3^{2001}-1=3^n-1\Leftrightarrow3^{2001}=3^n\Leftrightarrow n=2001\)