= Solution
The <single-block DNA zipper partition function> retains one contiguous paired block and ignores multiple separated bubbles. A block of $j\ge1$ paired bases can start at any of $N-j+1$ positions. Its first pair has initiation weight $\kappa$, and the remaining $j-1$ pairs each have propagation weight $s$. Thus
$$
\boxed{q=\kappa\sum_{j=1}^N(N-j+1)s^{j-1}.}
$$
The source's symbol $k$ in the displayed expressions denotes the same initiation constant as $\kappa$, not Boltzmann's constant. Association and other reference-state factors are absorbed into that statistical weight; the model treats the propagation weights as uniform despite the actual sequence's different bases.
Put $r=j-1$. Since $N-r$ counts how many terms of a finite <geometric series> contain $s^r$,
$$
\sum_{r=0}^{N-1}(N-r)s^r=\sum_{\ell=0}^{N-1}\sum_{r=0}^{\ell}s^r
=\frac1{s-1}\left[\frac{s(s^N-1)}{s-1}-N\right].
$$
Consequently
$$
\boxed{q=\frac{\kappa[s^{N+1}-(N+1)s+N]}{(s-1)^2}\quad(s\ne1).}
$$
The apparent singularity is removable: at $s=1$, the original sum gives $q=\kappa N(N+1)/2$, which is $45\kappa$ for these nine bases.
At $s\gg1$, the longest block dominates:
$$
\boxed{q\simeq\kappa s^{N-1}\left(1+\frac2s+\frac3{s^2}+\cdots\right),\qquad
q_{N=9}\simeq\kappa s^8.}
$$
Conditional on at least one pair, the fully paired <probability> approaches one, with leading correction of order $2/s$. If a fully unpaired reference state of weight one is included, the full <partition function> is $Q=1+q$, so a large propagation constant alone does not guarantee predominant duplex if $\kappa s^{N-1}$ is still very small.
The given sequence contains five A–T and four G–C pairs when fully complementary. Real duplex <stability> also depends strongly on stacking, sequence context, <salt> and <temperature>; it cannot be calculated just by counting two versus three <hydrogen bonds>. All else comparable, sequences with greater duplex propagation <stability> resist bubble opening more strongly, while easier-to-open, often A–T-rich start-site regions facilitate initiation. The homogeneous-$s$ zipper demonstrates that principle but does not resolve a quantitative sequence-specific melting profile or polymerase's active stabilization of the open state.
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