Sequence Pattern Analysis

The given sequence consists of strings made up of the letters 'A' and 'B'. Let's list the known terms and their lengths:

• Term 1: $A$ (Length: 1 = $2^0$)
• Term 2: $AB$ (Length: 2 = $2^1$)
• Term 3: $?$ (Expected Length: 4 = $2^2$)
• Term 4: $ABBABAAB$ (Length: 8 = $2^3$)

The lengths of the terms are increasing powers of 2.

Deriving the Missing Term

We can identify a recursive pattern. Let $S_n$ represent the $n$-th term in the sequence.

The Rule: Each subsequent term $S_{n+1}$ is formed by concatenating the previous term $S_n$ with its "complement". The complement of a string is obtained by replacing every 'A' with 'B' and every 'B' with 'A'. Mathematically, $S_{n+1} = S_n + \text{complement}(S_n)$.

1. Term 1: $S_1 = \text{'A'}$
2. Term 2: $S_2 = S_1 + \text{complement}(S_1) = \text{'A'} + \text{'B'} = \text{'AB'}$. This matches the second term given.
3. Term 3 (Missing Term): $S_3 = S_2 + \text{complement}(S_2) = \text{'AB'} + \text{complement}(\text{'AB'}) = \text{'AB'} + \text{'BA'} = \text{'ABBA'}$.
4. Term 4 (Verification): $S_4 = S_3 + \text{complement}(S_3) = \text{'ABBA'} + \text{complement}(\text{'ABBA'}) = \text{'ABBA'} + \text{'BAAB'} = \text{'ABBABAAB'}$. This matches the final term given.

The derived Term 3, $ABBA$, fits the pattern perfectly.

Thus, the missing word is ABBA.