1. Given a sequence of positive numbers $\left\{a_{n}\right\}$ satisfying $a_{n+1} \geqslant 2 a_{n}+1$, and $a_{n}<2^{n+1}$ for $n \in \mathbf{Z}_{+}$. Then the range of $a_{1}$ is
$\begin{array}{l}\text { i. } 1 .(0,3] . \\ \text { From } a_{n+1}+1 \geqslant 2\left(a_{n}+1\right) \\ \Rightarrow a_{n}+1 \geqslant\left(a_{1}+1\right) 2^{n-1} . \\ \text { Therefore }\left(a_{1}+1\right) 2^{n-1}-1 \leqslant a_{n}<2^{n+1} \\ \Rightarrow a_{1}+1<\frac{2^{n+1}+1}{2^{n-1}}=4+\frac{1}{2^{n-1}} \\ \Rightarrow a_{1}+1 \leqslant 4 \\ \Rightarrow 0<a_{1} \leqslant 3 .\end{array}$