= Solution
The <Dahlquist equivalence theorem> states that a consistent <linear multistep method> is convergent for suitably consistent starting values exactly when it is <zero-stable>. The <root condition for a multistep method> requires every root of $\rho$ to lie in the closed unit disk, with every unit-modulus root simple.
The roots are $1$ and $a$. Thus $|a|\leq1$ is necessary; $a=1$ is excluded because it gives a double root at one. At $a=-1$, the two unit roots are distinct, so the endpoint is allowed. Combined with part (a), this proves
$$
\boxed{-1\leq a<1\quad\text{for convergence}.}
$$
Assume a locally Lipschitz vector field, a smooth solution on the fixed time interval, a nearby solvable implicit branch and starting errors of the required order. The global order is three at $a=-1/5$ and two at the other convergent parameter values. Outside this interval, zero-step perturbations already grow through either an exterior root or a unit-root polynomial factor.
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