Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 4I / c

Regular. A DFA need only remember the current run length of $b$'s, using states $0,\dots ,99$ and a rejecting sink at 100.

Since $w_i=(ab{)}^{2^{i-1}}$, the language is $\{(ab{)}^{2^j}:j\ge 0\}$. If it were regular, pumping a sufficiently long member would alter its number of $ab$ blocks by a fixed positive amount, producing non-powers of two for some pumping exponent. Hence it is not regular.