Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 341 / 2 / b / Solution

Every member of the family has order at least three, so it is consistent. By the Dahlquist equivalence theorem, convergence is therefore equivalent to zero-stability, which is characterized by the root condition for a multistep method.

The roots of $\rho$ are $1$ and $a$. They lie in the closed unit disk exactly when $|a|\le 1$. At $a=1$, however, the unit root $1$ is repeated, whereas at $a=-1$ the distinct unit roots $1$ and $-1$ are both simple. Consequently