$8 \cdot 1$ Arrange all powers of 3 and the sums of any finite number of distinct powers of 3 in an increasing sequence:
$$1,3,4,9,10,12,13, \cdots$$

Find the 100th term of this sequence.
[Solution] Let the $n$-th term of the sequence be denoted as $a_{n}$. Clearly, $a_{n}$ is a positive integer, and its ternary representation uses only the digits 0 and 1. Conversely, any natural number whose ternary representation uses only the digits 0 and 1 must be a term in this sequence. Therefore, the binary representation of $n$ is exactly the ternary representation of $a_{n}$. Since the binary representation of 100 is 1100100, we have
$$a_{n}=3^{6}+3^{5}+3^{2}=981$$