[ Geometric progression $]$

Find the sum $1+2002+2002^{2}+\ldots+2002^{n}$.
Let's transform the expression $S=1+2002+2002^{2}+\ldots+2002^{n}=1+2002\left(S-2002^{n}\right)$. Solving this equation for $S$, we get $S=\frac{2002^{n+1}-1}{2001}$.