= Solution
In the imposed <free-draining polymer friction> model, the drag coefficients of the $N$ <monomers> add, giving $\zeta_1=N(6\pi\eta a)$ and $D_1=k_BT/(6\pi\eta aN)$. Taking the same three-dimensional centre-of-mass RMS convention as part (ii),
$$
\boxed{d_1=\sqrt{6D_1t}=\sqrt{\frac{k_BTt}{\pi\eta aN}}.}
$$
The <polymer> is a one-dimensional connected contour, but that alone does not restrict its <centre of mass> to one spatial axis. If the intended coordinate is instead literal one-dimensional curvilinear motion along an entanglement tube, its RMS contour displacement is $\sqrt{2D_1t}=\sqrt{k_BTt/(3\pi\eta aN)}$. That is a different coordinate and should not be compared with a three-dimensional displacement without stating the conversion/model.
Additive local drag is the free-draining <Rouse model> approximation. It neglects long-range solvent-mediated <hydrodynamic interactions> and is not a complete theory of an entangled chain. In a tube/<reptation> description, curvilinear <diffusion> can scale as $N^{-1}$ while long-time spatial centre-of-mass <diffusion> scales as $N^{-2}$ after the chain renews its tube. Conversely a dilute unentangled coil can exhibit <Zimm model> hydrodynamic coupling. <Monomer> motion, centre-of-mass motion and contour sliding therefore cannot all be identified with the same displacement law.
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