= Solution
By <orthonormality> in the <Generalized Bloch representation>,
$$
\operatorname{Tr}(\rho^2)=\frac1N+\sum_{k=1}^{N^2-1}s_k^2.
$$
A <pure state> has <density operator> $\rho=|\psi\rangle\langle\psi|$ with $\langle\psi|\psi\rangle=1$, so $\rho^2=\rho$ and its <purity of a density operator> is one. Consequently
$$
\boxed{\|s\|^2=1-\frac1N.}
$$
Every <pure state> therefore maps to the specified radius-$\sqrt{1-1/N}$ <sphere> in $\mathbb R^{N^2-1}$, of dimension $N^2-2$. The expansion is unique, so it distinguishes pure-state <density operators>, while vectors differing only by <global phase> describe the same point. This proves that pure states lie on the sphere, without asserting that they fill it.
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