= Solution
Insert the exact data $y(t)=P(t)$ into the <multistep method>[linear multistep method]. Its local residual is the linear functional
$$
\mathcal L_h[P]
=\sum_{k=0}^s\rho_kP(t_{n+k})
-h\sum_{k=0}^s\sigma_kP'(t_{n+k}).
$$
Taylor expansion about $t_n$ shows that a method has order $p$ exactly when this functional annihilates the monomials $1,t,\ldots,t^p$; by linearity this is equivalent to annihilating every polynomial of degree at most $p$. This is precisely
$$
\boxed{
\sum_{k=0}^s\rho_kP(t_{n+k})
=h\sum_{k=0}^s\sigma_kP'(t_{n+k})}.
$$
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