Example 4 The increasing sequence $1,3,4,9,10,12,13, \cdots$ consists of some positive integers, which are either powers of 3 or sums of several different powers of 3. Find the 100th term of this sequence.
Let the sequence be denoted as $\left\{a_{n}\right\}$. By the problem, we think of representing the known terms using ternary (base-3) notation. They are sequentially $a_{1}=1, a_{2}=10, a_{3}=11, a_{4}=100, a_{5}=101, a_{6}=110, a_{7}=111$. These numbers are clearly not consecutive positive integers, but if we formally treat the above ternary representations as binary (base-2) representations, they exactly represent the consecutive positive integers from 1 to 7. Therefore, $a_{100}$ corresponds to the binary representation 1100100 (which is the positive integer 100 in decimal). Thus, treating 1100100 as a ternary number, we have $a_{100}=$ $(1100100)_{3}=3^{6}+3^{5}+3^{2}=981$.