= Solution
For an <outer solution> with bounded derivatives as $\epsilon\to0$, the final complex equation gives $r_n=0$. Working backwards in $0=r_i-r_{i+1}$ then gives every $r_i=0$, including $r_1$; setting only their sum to zero would not suffice. This is the <quasi-steady-state approximation>, valid after the initial complex-formation layer with the relevant dimensionless rate parameters held fixed.
The recurrence is $C_i=SC_{i-1}/K_i$, where $K_i=(k_{-i}+k_i^+)/k_i$. Thus $C_j=\phi_jS^jC_0$, $\phi_j=\prod_{i=1}^jK_i^{-1}$. Substitution into <enzyme conservation> yields
$$
C_0=\frac{E_0}{1+\sum_{j=1}^n\phi_jS^j},\qquad C_j=\frac{E_0\phi_jS^j}{1+\sum_{\ell=1}^n\phi_\ell S^\ell}.
$$
Each state $j$ produces <reaction product> at rate $k_j^+C_j$, so
$$
\boxed{\rho=E_0\frac{\sum_{j=1}^nk_j^+\phi_jS^j}{1+\sum_{j=1}^n\phi_jS^j}.}
$$
The source's uppercase $N$ in the index range is read as the number $n$ of binding sites. This approximation preserves the total <enzyme> population and the different catalytic rates of the occupied states.
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