The increasing sequence $1; 3; 4; 9; 10; 12; 13; 27; 28; 30; 31, \ldots$ is formed with positive integers which are powers of $3$ or sums of different powers of $3$. Which number is in the $100^{th}$ position?
1. **Understanding the Sequence**:
   The sequence is formed by positive integers which are either powers of 3 or sums of different powers of 3. This means each number in the sequence can be represented as a sum of distinct powers of 3. For example:
   - \(1 = 3^0\)
   - \(3 = 3^1\)
   - \(4 = 3^1 + 3^0\)
   - \(9 = 3^2\)
   - \(10 = 3^2 + 3^0\)
   - and so on.

2. **Binary Representation Analogy**:
   Each number in the sequence can be thought of as a binary number where each digit represents whether a particular power of 3 is included (1) or not (0). For example:
   - \(4 = 3^1 + 3^0\) can be represented as \(11_3\) (in base 3).
   - \(10 = 3^2 + 3^0\) can be represented as \(101_3\) (in base 3).

3. **Finding the 100th Term**:
   To find the 100th term, we need to consider the binary representation of 100 and then interpret it in base 3. The binary representation of 100 is \(1100100_2\).

4. **Converting Binary to Base 3**:
   We interpret the binary number \(1100100_2\) as a base 3 number:
   \[
   1100100_3 = 1 \cdot 3^6 + 1 \cdot 3^5 + 0 \cdot 3^4 + 0 \cdot 3^3 + 1 \cdot 3^2 + 0 \cdot 3^1 + 0 \cdot 3^0
   \]
   Simplifying this, we get:
   \[
   1100100_3 = 3^6 + 3^5 + 3^2
   \]

5. **Calculating the Value**:
   Now, we calculate the value of each term:
   \[
   3^6 = 729
   \]
   \[
   3^5 = 243
   \]
   \[
   3^2 = 9
   \]
   Adding these together:
   \[
   3^6 + 3^5 + 3^2 = 729 + 243 + 9 = 981
   \]

Conclusion:
\[
\boxed{981}
\]