The increasing sequence $1,3,4,9,10,12,13\cdots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the $100^{\text{th}}$ term of this sequence.
1. **Identify the sequence in base 3:**
   The given sequence consists of numbers that are either powers of 3 or sums of distinct powers of 3. The first few terms are \(1, 3, 4, 9, 10, 12, 13, \ldots\). When these numbers are written in base 3, they become \(1_3, 10_3, 11_3, 100_3, 101_3, 110_3, 111_3, \ldots\).

2. **Establish the correspondence:**
   Notice that the sequence in base 3 corresponds to the sequence of all binary numbers \(1_2, 10_2, 11_2, 100_2, 101_2, 110_2, 111_2, \ldots\). This means that the \(n\)-th term of the sequence can be found by interpreting \(n\) in binary and then converting it to base 3.

3. **Find the 100th term:**
   To find the 100th term, we first convert 100 to binary:
   \[
   100_{10} = 1100100_2
   \]

4. **Convert the binary number to base 3:**
   Now, we interpret \(1100100_2\) as a base 3 number:
   \[
   1100100_3 = 1 \cdot 3^6 + 1 \cdot 3^5 + 0 \cdot 3^4 + 0 \cdot 3^3 + 1 \cdot 3^2 + 0 \cdot 3^1 + 0 \cdot 3^0
   \]
   Calculate each term:
   \[
   1 \cdot 3^6 = 729
   \]
   \[
   1 \cdot 3^5 = 243
   \]
   \[
   1 \cdot 3^2 = 9
   \]
   Sum these values:
   \[
   729 + 243 + 9 = 981
   \]

Conclusion:
\[
\boxed{981}
\]