9. (16 points) Let the increasing sequence $a_{1}, a_{2}, \cdots, a_{n}, \cdots$ satisfy $a_{n}+a_{n+1}=2^{n}$ for $n \in \mathbf{Z}_{+}$. Find the range of values for $a_{1}$.


9.  $a_{1}+a_{2}=2\left(a_{1}0 \text {. }
$$

Thus, $0<a_{1}<1$.
Taking $0<a_{1}<1$, then $1<a_{2}<2$.
Assume that
$$
2^{n-2}<a_{n}<2^{n-1}(n \geqslant 2) \text {. }
$$

From $a_{n}+a_{n+1}=2^{n}$
$$
\begin{array}{l}
\Rightarrow 2^{n}-2^{n-1}<a_{n+1}<2^{n}-2^{n-2} \\
\Rightarrow 2^{n-1}<a_{n+1}<2^{n} .
\end{array}
$$

Therefore, for $n=2,3, \cdots$, we have
$$
2^{n-2}<a_{n}<2^{n-1} \text {. }
$$

Thus, $a_{1}<a_{2}<\cdots$ is an increasing sequence, satisfying the requirements. Hence, $a_{1} \in(0,1)$.