A king puts 1 grain of rice on the first square of a board, 2 on the second, 4 on the third, and so…

The sum of the first $n$ terms is $S_n = \dfrac{a(r^n - 1)}{r - 1}$ for $r \ne 1$, and $na$ for $r = 1$. When $|r| < 1$, the sum to infinity is $S_\infty = \dfrac{a}{1 - r}$. A recurring decimal is a GP with ratio $\tfrac{1}{10}$, $\tfrac{1}{100}$ and so on.