= Solution
The <Dahlquist equivalence theorem> says that a consistent linear multistep method is convergent exactly when it is zero-stable. Here
$$
\rho(z)=z^3+(2a-3)z^2-(2a-3)z-1
=(z-1)\bigl(z^2+2(a-1)z+1\bigr).
$$
The consistency conditions hold for every $a$. The two quadratic roots have product one. They are distinct and on the unit circle exactly when $0<a<2$; at $a=2$ the root $-1$ is repeated, while outside this interval one root has modulus greater than one. The <root condition for a multistep method> therefore gives
$$
\boxed{\text{convergence exactly for }0<a<2}.
$$
The polynomial exactness conditions hold through degree two, while the degree-three residual is $6-a$. Every convergent member thus has
$$
\boxed{\text{order }2}.
$$
(The exceptional value $a=6$ has formal order four but is not zero-stable.)
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