  Let ${\cal O}_n$ denote the Cuntz algebra for $n\geq 2$. We introduce an
embedding $f$ of ${\cal O}_m$ into ${\cal O}_n$ arising from a geometric
progression of Cuntz generaters of ${\cal O}_n$. By identifying ${\cal O}_m$
with $f({\cal O}_m)$, we extend Cuntz states on ${\cal O}_m$ to ${\cal O}_n$.
We show (i) a necessary and sufficient condition of the uniqueness of the
extension, (ii) the complete classification of all such extensions up to
unitary equivalence of their GNS representations, and (iii) the decomposition
formula of a mixing state into a convex hull of pure states. The complete set
of invariants of all GNS representations by such pure states is given as a
certain set of complex unit vectors.
