= Solution
Let $\theta_j=j\pi/N$ and retain the positive <spin> beam at every stage. The overlap of successive positive <eigenstates> is
$$
\langle\psi_{\theta_j}^+|\psi_{\theta_{j-1}}^+\rangle
=\cos\!\left(\frac{\pi}{2N}\right).
$$
Thus the unconditional survival <probability>, relative to the original intensity, is
$$
\boxed{p_N=\left[\cos^2\!\left(\frac{\pi}{2N}\right)\right]^N.}
$$
The last retained <quantum state> is $|\psi_\pi^+\rangle=|\downarrow\rangle$, so its final negative-$z$ result is certain conditional on survival and adds no further factor. In particular $p_2=1/4$, $p_{10}\simeq0.7805$, and $p_{100}\simeq0.9756$. Since $\log\cos x=-x^2/2+O(x^4)$,
$$
\log p_N=-\frac{\pi^2}{4N}+O(N^{-3}),\qquad \boxed{\lim_{N\to\infty}p_N=1.}
$$
This is a changing-measurement-axis form of the <quantum Zeno effect>: frequent selections steer the surviving <spin> through a continuous succession of nearby <eigenstates>.
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