Consider the sequence $1,3,4,9,10,12,13 \ldots$ consisting of integers greater than or equal to 1, in ascending order, which are powers of 3 or sums of distinct powers of 3 (for example: $4=3^{1}+3^{0}, 10=3^{2}+3^{0}, 13=3^{2}+3^{1}+3^{0} \ldots$ ).

What is the integer that appears at the hundredth position?
For this exercise, we need to think in base 3 and base 2. The sequence consists of numbers whose base 3 representation contains only 1s and 0s (in the sequence, I will write the base 2 and base 3 representations in the form $\overline{11000}^{2}$ and $\overline{1101}^{3}$): $4=3^{1}+3^{0}=\overline{11}^{3}, 10=3^{2}+3^{0}=$ $\overline{101}^{3}, 13=3^{2}+3^{1}+3^{0}=\overline{111}^{3}, \ldots$ If we write all the numbers in the list in base 3, we will see all the numbers that can be written with 0s and 1s. But the numbers that can be written only with 0s and 1s correspond to the binary representations of all integers. We need to determine the 100th of these numbers: it is the one that corresponds to the binary representation of 100.

$$
\begin{gathered}
100=64+32+4=\overline{1100100}^{2} \\
\overline{1100100}^{3}=3^{6}+3^{5}+3^{2}=729+243+9=981
\end{gathered}
$$

The 100th number in the list is 981.