= Solution
Let $\beta=(k_BT)^{-1}$, and sum over all $2^N$ configurations satisfying the periodic identification. The <canonical partition function> and finite-size <free energy> per site are
$$
\boxed{Z_N=\sum_{\sigma_1,\ldots,\sigma_N=\pm1}
\exp\!\left[-\beta J\sum_j\sigma_j\sigma_{j+1}+\beta B\sum_j\sigma_j\right],\qquad
f_N=-\frac{1}{\beta N}\log Z_N.}
$$
The plus sign multiplying $J$ in the energy means that $J>0$ favors opposite neighboring spins: this is an <antiferromagnetic Ising model>. Even $N$ permits an alternating configuration to close without frustration. The finite-size formula does not require taking a <thermodynamic limit>; that limit is taken only after evaluating $Z_N$.
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